Topic Quiz • 146 questions • 40 seconds per question.
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If \(\theta=0^\circ\), what is the value of \(\sin \theta\)?
C. (0)
\(\sin 0^\circ=0\) होता है। परीक्षा में मुख्य कोणों के मान याद रखें। / \(\sin 0^\circ=0\). In exams, remember the values of standard angles.
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If \(\theta=0^\circ\), what is the value of \(\cos \theta\)?
A. (1)
\(\cos 0^\circ=1\) होता है। आधारभूत कोणों की सारणी से उत्तर जल्दी मिलता है। / \(\cos 0^\circ=1\). The table of standard angles helps you answer quickly.
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If \(\theta=45^\circ\), what is the value of \(\tan \theta\)?
D. (1)
\(\tan 45^\circ=1\) होता है। \(\tan \theta\) के मुख्य मान अलग से याद रखें। / \(\tan 45^\circ=1\). Learn the standard values of \(\tan \theta\) separately.
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Which trigonometric function is the reciprocal of \(\sin \theta\)?
B. \(\cosec \theta\)
\(\cosec \theta=\frac{1}{\sin \theta}\) होता है। व्युत्क्रम संबंधों को जोड़ों में याद करें। / \(\cosec \theta=\frac{1}{\sin \theta}\). Remember reciprocal relations in pairs.
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Which trigonometric function is the reciprocal of \(\cos \theta\)?
B. \(\sec \theta\)
\(\sec \theta=\frac{1}{\cos \theta}\) होता है। परीक्षा में \(\sec \theta\) और \(\cosec \theta\) को न मिलाएँ। / \(\sec \theta=\frac{1}{\cos \theta}\). Do not confuse \(\sec \theta\) with \(\cosec \theta\) in exams.
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Which trigonometric function is the reciprocal of \(\tan \theta\)?
C. \(\cot \theta\)
\(\cot \theta=\frac{1}{\tan \theta}\) होता है। व्युत्क्रम फलनों को सूत्र रूप में लिखकर जाँचें। / \(\cot \theta=\frac{1}{\tan \theta}\). Check reciprocal functions by writing them as formulas.
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What is \(\tan \theta\) equal to?
A. \(\frac{\sin \theta}{\cos \theta}\)
\(\tan \theta=\frac{\sin \theta}{\cos \theta}\) होता है। भाग वाले संबंध पहचानना आसान प्रश्नों में उपयोगी है। / \(\tan \theta=\frac{\sin \theta}{\cos \theta}\). Recognizing quotient relations is useful in easy questions.
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What is \(\cot \theta\) equal to?
D. \(\frac{\cos \theta}{\sin \theta}\)
\(\cot \theta=\frac{\cos \theta}{\sin \theta}\) होता है। \(\tan \theta\) और \(\cot \theta\) के संबंध उलटे होते हैं। / \(\cot \theta=\frac{\cos \theta}{\sin \theta}\). The relations of \(\tan \theta\) and \(\cot \theta\) are reciprocal.
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What is the value of the basic identity \(\sin^2 \theta+\cos^2 \theta\)?
C. (1)
\(\sin^2 \theta+\cos^2 \theta=1\) मूल सर्वसमिका है। इसे हर अध्याय में आधार सूत्र की तरह प्रयोग करें। / \(\sin^2 \theta+\cos^2 \theta=1\) is a basic identity. Use it as a base formula throughout the chapter.
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What is the value of \(\sec^2 \theta-\tan^2 \theta\)?
D. (1)
\(\sec^2 \theta=1+\tan^2 \theta\) से मान (1) मिलता है। सूत्र को स्थानांतरित करके उत्तर निकालें। / From \(\sec^2 \theta=1+\tan^2 \theta\), the value is (1). Rearrange the formula to get the answer.
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What is the fundamental period of \(\sin \theta\)?
B. \(2\pi\)
\(\sin \theta\) का मूल काल \(2\pi\) होता है। काल संबंधी प्रश्नों में फलन का नाम पहले पहचानें। / The fundamental period of \(\sin \theta\) is \(2\pi\). In period questions, identify the function first.
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What is the fundamental period of \(\cos \theta\)?
D. \(2\pi\)
\(\cos \theta\) का मूल काल \(2\pi\) होता है। \(\sin \theta\) और \(\cos \theta\) का काल समान है। / The fundamental period of \(\cos \theta\) is \(2\pi\). \(\sin \theta\) and \(\cos \theta\) have the same period.
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What is the fundamental period of \(\tan \theta\)?
C. \(\pi\)
\(\tan \theta\) का मूल काल \(\pi\) होता है। \(\tan \theta\) का काल \(\sin \theta\) से अलग है। / The fundamental period of \(\tan \theta\) is \(\pi\). The period of \(\tan \theta\) differs from that of \(\sin \theta\).
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What is the fundamental period of \(\cot \theta\)?
A. \(\pi\)
\(\cot \theta\) का मूल काल \(\pi\) होता है। \(\tan \theta\) और \(\cot \theta\) का काल समान है। / The fundamental period of \(\cot \theta\) is \(\pi\). \(\tan \theta\) and \(\cot \theta\) have the same period.
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What is (\cosec\(-\theta\)) equal to?
B. \(-\cosec \theta\)
\(\cosec \theta\) विषम फलन है इसलिए (\cosec\(-\theta\)=-\cosec \theta)। यह \(\sin \theta\) के व्युत्क्रम जैसा व्यवहार करता है। / \(\cosec \theta\) is an odd function, so (\cosec\(-\theta\)=-\cosec \theta). It behaves like the reciprocal of \(\sin \theta\).
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What is the range of \(\sin \theta\)?
D. ([-1,1])
\(\sin \theta\) का मान हमेशा ([-1,1]) में रहता है। परिसर पूछे जाने पर अधिकतम और न्यूनतम मान देखें। / The value of \(\sin \theta\) always lies in ([-1,1]). For range questions, check the maximum and minimum values.
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Which is the range of \(\cos \theta\)?
A. ([-1,1])
\(\cos \theta\) का परिसर ([-1,1]) है। \(\sin \theta\) और \(\cos \theta\) का परिसर समान होता है। / The range of \(\cos \theta\) is ([-1,1]). \(\sin \theta\) and \(\cos \theta\) have the same range.
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What is the range of \(\tan \theta\)?
C. (\(-\infty,\infty\))
\(\tan \theta\) सभी वास्तविक मान ले सकता है। इसलिए इसका परिसर (\(-\infty,\infty\)) है। / \(\tan \theta\) can take all real values. Therefore its range is (\(-\infty,\infty\)).
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Which is the range of \(\sec \theta\)?
B. (\(-\infty,-1]\cup[1,\infty\))
\(\sec \theta=\frac{1}{\cos \theta}\) होने से इसके मान ([-1,1]) के अंदर नहीं आते। परिसर (\(-\infty,-1]\cup[1,\infty\)) होता है। / Since \(\sec \theta=\frac{1}{\cos \theta}\), its values do not lie inside ([-1,1]). Its range is (\(-\infty,-1]\cup[1,\infty\)).
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What is the range of \(\cosec \theta\)?
D. (\(-\infty,-1]\cup[1,\infty\))
\(\cosec \theta=\frac{1}{\sin \theta}\) होता है। इसका परिसर (\(-\infty,-1]\cup[1,\infty\)) है। / \(\cosec \theta=\frac{1}{\sin \theta}\). Its range is (\(-\infty,-1]\cup[1,\infty\)).
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What is the range of \(\cot \theta\)?
A. (\(-\infty,\infty\))
\(\cot \theta\) सभी वास्तविक मान ले सकता है। \(\tan \theta\) और \(\cot \theta\) का परिसर समान है। / \(\cot \theta\) can take all real values. \(\tan \theta\) and \(\cot \theta\) have the same range.
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When is \(\tan \theta\) undefined?
B. जब \(\cos \theta=0\)when \(\cos \theta=0\)
\(\tan \theta=\frac{\sin \theta}{\cos \theta}\) है इसलिए \(\cos \theta=0\) पर यह अपरिभाषित होता है। हर में शून्य नहीं होना चाहिए। / \(\tan \theta=\frac{\sin \theta}{\cos \theta}\), so it is undefined when \(\cos \theta=0\). The denominator must not be zero.
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When is \(\sec \theta\) undefined?
A. जब \(\cos \theta=0\)when \(\cos \theta=0\)
\(\sec \theta=\frac{1}{\cos \theta}\) है इसलिए \(\cos \theta=0\) पर यह अपरिभाषित है। व्युत्क्रम फलन में हर की जाँच करें। / \(\sec \theta=\frac{1}{\cos \theta}\), so it is undefined when \(\cos \theta=0\). Check the denominator in reciprocal functions.
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When is \(\cosec \theta\) undefined?
C. जब \(\sin \theta=0\)when \(\sin \theta=0\)
\(\cosec \theta=\frac{1}{\sin \theta}\) है इसलिए \(\sin \theta=0\) पर यह अपरिभाषित होता है। हर शून्य होने पर फलन नहीं बनता। / \(\cosec \theta=\frac{1}{\sin \theta}\), so it is undefined when \(\sin \theta=0\). A function is not defined when the denominator is zero.
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When is \(\cot \theta\) undefined?
D. जब \(\sin \theta=0\)when \(\sin \theta=0\)
\(\cot \theta=\frac{\cos \theta}{\sin \theta}\) है इसलिए \(\sin \theta=0\) पर यह अपरिभाषित होता है। भाग वाले फलनों में हर पहचानें। / \(\cot \theta=\frac{\cos \theta}{\sin \theta}\), so it is undefined when \(\sin \theta=0\). In quotient functions, identify the denominator.
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What is (\sin\(\pi+\theta\)) equal to?
A. -\(\sin \theta\)
(\sin\(\pi+\theta\)=-\sin \theta) होता है। तीसरे चतुर्थांश में \(\sin \theta\) का चिह्न ऋण होता है। / (\sin\(\pi+\theta\)=-\sin \theta). In the third quadrant, \(\sin \theta\) is negative.
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What is (\cos\(\pi+\theta\)) equal to?
B. -\(\cos \theta\)
(\cos\(\pi+\theta\)=-\cos \theta) होता है। तीसरे चतुर्थांश में \(\cos \theta\) का चिह्न ऋण होता है। / (\cos\(\pi+\theta\)=-\cos \theta). In the third quadrant, \(\cos \theta\) is negative.
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What is (\tan\(\pi+\theta\)) equal to?
C. \(\tan \theta\)
\(\tan \theta\) का काल \(\pi\) है इसलिए (\tan\(\pi+\theta\)=\tan \theta)। काल याद रखने से रूपांतरण आसान होता है। / The period of \(\tan \theta\) is \(\pi\), so (\tan\(\pi+\theta\)=\tan \theta). Knowing periods makes transformations easier.
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What is (\sin\(2\pi+\theta\)) equal to?
D. \(\sin \theta\)
\(\sin \theta\) का काल \(2\pi\) है इसलिए मान \(\sin \theta\) ही रहता है। पूर्ण चक्र जुड़ने पर \(\sin \theta\) नहीं बदलता। / The period of \(\sin \theta\) is \(2\pi\), so the value remains \(\sin \theta\). Adding a full cycle does not change \(\sin \theta\).
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What is (\cos\(2\pi+\theta\)) equal to?
A. \(\cos \theta\)
\(\cos \theta\) का काल \(2\pi\) है। इसलिए (\cos\(2\pi+\theta\)=\cos \theta) होता है। / The period of \(\cos \theta\) is \(2\pi\). Therefore (\cos\(2\pi+\theta\)=\cos \theta).
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What is (\sin\(\pi-\theta\)) equal to?
B. \(\sin \theta\)
(\sin\(\pi-\theta\)=\sin \theta) होता है। दूसरे चतुर्थांश में \(\sin \theta\) धनात्मक होता है। / (\sin\(\pi-\theta\)=\sin \theta). In the second quadrant, \(\sin \theta\) is positive.
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What is (\cos\(\pi-\theta\)) equal to?
C. -\(\cos \theta\)
(\cos\(\pi-\theta\)=-\cos \theta) होता है। दूसरे चतुर्थांश में \(\cos \theta\) ऋणात्मक होता है। / (\cos\(\pi-\theta\)=-\cos \theta). In the second quadrant, \(\cos \theta\) is negative.
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What is (\tan\(\pi-\theta\)) equal to?
D. -\(\tan \theta\)
(\tan\(\pi-\theta\)=-\tan \theta) होता है। दूसरे चतुर्थांश में \(\tan \theta\) ऋणात्मक होता है। / (\tan\(\pi-\theta\)=-\tan \theta). In the second quadrant, \(\tan \theta\) is negative.
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What is (\sin\left\(\frac{\pi}{2}-\theta\right\)) equal to?
A. \(\cos \theta\)
(\sin\left\(\frac{\pi}{2}-\theta\right\)=\cos \theta) सह-फलन संबंध है। \(\frac{\pi}{2}\) वाले रूपांतरण में फलन बदलता है। / (\sin\left\(\frac{\pi}{2}-\theta\right\)=\cos \theta) is a cofunction relation. In transformations with \(\frac{\pi}{2}\), the function changes.
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What is (\cos\left\(\frac{\pi}{2}-\theta\right\)) equal to?
B. \(\sin \theta\)
(\cos\left\(\frac{\pi}{2}-\theta\right\)=\sin \theta) होता है। सह-फलन संबंध में \(\sin \theta\) और \(\cos \theta\) आपस में बदलते हैं। / (\cos\left\(\frac{\pi}{2}-\theta\right\)=\sin \theta). In cofunction relations, \(\sin \theta\) and \(\cos \theta\) interchange.
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What is (\tan\left\(\frac{\pi}{2}-\theta\right\)) equal to?
C. \(\cot \theta\)
(\tan\left\(\frac{\pi}{2}-\theta\right\)=\cot \theta) होता है। \(\tan \theta\) का सह-फलन \(\cot \theta\) है। / (\tan\left\(\frac{\pi}{2}-\theta\right\)=\cot \theta). The cofunction of \(\tan \theta\) is \(\cot \theta\).
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What is (\sec\left\(\frac{\pi}{2}-\theta\right\)) equal to?
D. \(\cosec \theta\)
(\sec\left\(\frac{\pi}{2}-\theta\right\)=\cosec \theta) होता है। \(\sec \theta\) और \(\cosec \theta\) सह-फलन हैं। / (\sec\left\(\frac{\pi}{2}-\theta\right\)=\cosec \theta). \(\sec \theta\) and \(\cosec \theta\) are cofunctions.
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What is the value of \(\sin 30^\circ\)?
A. \(\frac{1}{2}\)
\(\sin 30^\circ=\frac{1}{2}\) होता है। \(30^\circ\) और \(60^\circ\) के मान अक्सर पूछे जाते हैं। / \(\sin 30^\circ=\frac{1}{2}\). Values of \(30^\circ\) and \(60^\circ\) are often asked.
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What is the value of \(\cos 60^\circ\)?
B. \(\frac{1}{2}\)
\(\cos 60^\circ=\frac{1}{2}\) होता है। \(\sin 30^\circ\) और \(\cos 60^\circ\) बराबर होते हैं। / \(\cos 60^\circ=\frac{1}{2}\). \(\sin 30^\circ\) and \(\cos 60^\circ\) are equal.
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What is the value of \(\tan 30^\circ\)?
C. \(\frac{1}{\sqrt{3}}\)
\(\tan 30^\circ=\frac{1}{\sqrt{3}}\) होता है। \(\tan 30^\circ\) और \(\tan 60^\circ\) को अलग-अलग याद करें। / \(\tan 30^\circ=\frac{1}{\sqrt{3}}\). Remember \(\tan 30^\circ\) and \(\tan 60^\circ\) separately.
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What is the value of \(\sin 90^\circ\)?
D. (1)
\(\sin 90^\circ=1\) होता है। मुख्य कोणों की तालिका से यह सीधा मान मिलता है। / \(\sin 90^\circ=1\). This value comes directly from the standard-angle table.
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What is the value of \(\cos 90^\circ\)?
A. (0)
\(\cos 90^\circ=0\) होता है। \(90^\circ\) पर \(\cos \theta\) शून्य हो जाता है। / \(\cos 90^\circ=0\). At \(90^\circ\), \(\cos \theta\) becomes zero.
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What is the value of \(\tan 0^\circ\)?
B. (0)
\(\tan 0^\circ=0\) होता है। \(\tan \theta=\frac{\sin \theta}{\cos \theta}\) से भी इसे जाँचा जा सकता है। / \(\tan 0^\circ=0\). It can also be checked using \(\tan \theta=\frac{\sin \theta}{\cos \theta}\).
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What is the value of \(\sec 0^\circ\)?
C. (1)
\(\sec 0^\circ=\frac{1}{\cos 0^\circ}=1\) होता है। व्युत्क्रम मान निकालते समय मूल फलन का मान पहले लिखें। / \(\sec 0^\circ=\frac{1}{\cos 0^\circ}=1\). While finding reciprocal values, first write the value of the basic function.
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What can \(\sin^2 \theta\) be written as?
A. \(1-\cos^2 \theta\)
\(\sin^2 \theta+\cos^2 \theta=1\) से \(\sin^2 \theta=1-\cos^2 \theta\) मिलता है। परीक्षा में मूल सर्वसमिका को पहले लिखें। / From \(\sin^2 \theta+\cos^2 \theta=1\), we get \(\sin^2 \theta=1-\cos^2 \theta\). In exams, write the basic identity first.
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What is \(\tan^2 \theta+1\) equal to?
B. \(\sec^2 \theta\)
\(\sec^2 \theta=1+\tan^2 \theta\) होता है। \(\tan \theta\) वाले प्रश्नों में \(\sec^2 \theta\) पहचानना उपयोगी है। / \(\sec^2 \theta=1+\tan^2 \theta\). In questions involving \(\tan \theta\), recognizing \(\sec^2 \theta\) is useful.
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What is the sign of \(\sin \theta\) in the first quadrant?
A. धनात्मकPositive
पहले चतुर्थांश में सभी त्रिकोणमितीय फलन धनात्मक होते हैं। परीक्षा में चतुर्थांश के चिह्न याद रखें। / In the first quadrant, all trigonometric functions are positive. Remember quadrant signs for exams.
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Which function is positive in the second quadrant?
B. \(\sin \theta\)
दूसरे चतुर्थांश में \(\sin \theta\) और \(\cosec \theta\) धनात्मक होते हैं। चिह्न नियम को याद रखने से ऐसे प्रश्न जल्दी हल होते हैं। / In the second quadrant, \(\sin \theta\) and \(\cosec \theta\) are positive. Knowing sign rules helps solve such questions quickly.
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What is the sign of \(\tan \theta\) in the third quadrant?
C. धनात्मकPositive
तीसरे चतुर्थांश में \(\tan \theta\) और \(\cot \theta\) धनात्मक होते हैं। चतुर्थांश के अनुसार चिह्न बदलता है। / In the third quadrant, \(\tan \theta\) and \(\cot \theta\) are positive. The sign changes according to the quadrant.
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Which function is positive in the fourth quadrant?
D. \(\cos \theta\)
चौथे चतुर्थांश में \(\cos \theta\) और \(\sec \theta\) धनात्मक होते हैं। संकेत नियम को छोटी तालिका से याद करें। / In the fourth quadrant, \(\cos \theta\) and \(\sec \theta\) are positive. Remember sign rules using a small table.
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What is (\sin\left\(\frac{\pi}{2}+\theta\right\)) equal to?
A. \(\cos \theta\)
\(\frac{\pi}{2}\) के साथ \(\sin \theta\) सह-फलन \(\cos \theta\) में बदलता है। दूसरे चतुर्थांश में चिह्न धनात्मक रहता है। / With \(\frac{\pi}{2}\), \(\sin \theta\) changes to its cofunction \(\cos \theta\). The sign remains positive in the second quadrant.
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What is (\cos\left\(\frac{\pi}{2}+\theta\right\)) equal to?
B. -\(\sin \theta\)
(\cos\left\(\frac{\pi}{2}+\theta\right\)=-\sin \theta) होता है। \(\frac{\pi}{2}\) पर फलन बदलता है और चिह्न चतुर्थांश से तय होता है। / (\cos\left\(\frac{\pi}{2}+\theta\right\)=-\sin \theta). At \(\frac{\pi}{2}\), the function changes and the sign is decided by the quadrant.
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What is (\tan\left\(\frac{\pi}{2}+\theta\right\)) equal to?
C. -\(\cot \theta\)
\(\tan \theta\) का सह-फलन \(\cot \theta\) है और दूसरे चतुर्थांश में चिह्न ऋणात्मक होता है। इसलिए उत्तर \(-\cot \theta\) है। / The cofunction of \(\tan \theta\) is \(\cot \theta\), and the sign is negative in the second quadrant. Therefore the answer is \(-\cot \theta\).
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What is (\cot\left\(\frac{\pi}{2}+\theta\right\)) equal to?
D. -\(\tan \theta\)
\(\cot \theta\) का सह-फलन \(\tan \theta\) है और दूसरे चतुर्थांश में चिह्न ऋणात्मक होता है। ऐसे प्रश्नों में पहले सह-फलन पहचानें। / The cofunction of \(\cot \theta\) is \(\tan \theta\), and the sign is negative in the second quadrant. In such questions, first identify the cofunction.
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What is (\sec\left\(\frac{\pi}{2}+\theta\right\)) equal to?
A. -\(\cosec \theta\)
\(\sec \theta\) का सह-फलन \(\cosec \theta\) है और दूसरे चतुर्थांश में \(\sec \theta\) ऋणात्मक होता है। इसलिए मान \(-\cosec \theta\) है। / The cofunction of \(\sec \theta\) is \(\cosec \theta\), and \(\sec \theta\) is negative in the second quadrant. Hence the value is \(-\cosec \theta\).
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What is (\cosec\left\(\frac{\pi}{2}+\theta\right\)) equal to?
B. \(\sec \theta\)
\(\cosec \theta\) का सह-फलन \(\sec \theta\) है और दूसरे चतुर्थांश में चिह्न धनात्मक रहता है। सह-फलन जोड़ों को याद रखें। / The cofunction of \(\cosec \theta\) is \(\sec \theta\), and the sign remains positive in the second quadrant. Remember cofunction pairs.
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What is (\sin\left\(\frac{3\pi}{2}-\theta\right\)) equal to?
C. -\(\cos \theta\)
\(\frac{3\pi}{2}\) के साथ \(\sin \theta\) सह-फलन में बदलता है और तीसरे चतुर्थांश में चिह्न ऋणात्मक होता है। इसलिए उत्तर \(-\cos \theta\) है। / With \(\frac{3\pi}{2}\), \(\sin \theta\) changes to a cofunction and the sign is negative in the third quadrant. So the answer is \(-\cos \theta\).
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What is (\cos\left\(\frac{3\pi}{2}-\theta\right\)) equal to?
A. -\(\sin \theta\)
\(\frac{3\pi}{2}\) के साथ \(\cos \theta\) सह-फलन \(\sin \theta\) में बदलता है। तीसरे चतुर्थांश में \(\cos \theta\) ऋणात्मक होता है। / With \(\frac{3\pi}{2}\), \(\cos \theta\) changes to the cofunction \(\sin \theta\). In the third quadrant, \(\cos \theta\) is negative.
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What is (\tan\left\(\frac{3\pi}{2}-\theta\right\)) equal to?
B. \(\cot \theta\)
\(\frac{3\pi}{2}\) के साथ \(\tan \theta\) \(\cot \theta\) में बदलता है। तीसरे चतुर्थांश में \(\tan \theta\) धनात्मक होता है। / With \(\frac{3\pi}{2}\), \(\tan \theta\) changes to \(\cot \theta\). In the third quadrant, \(\tan \theta\) is positive.
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What is (\sin\left\(\frac{3\pi}{2}+\theta\right\)) equal to?
C. -\(\cos \theta\)
(\sin\left\(\frac{3\pi}{2}+\theta\right\)=-\cos \theta) होता है। \(\frac{3\pi}{2}\) वाले रूप में \(\sin \theta\) सह-फलन में बदलता है। / (\sin\left\(\frac{3\pi}{2}+\theta\right\)=-\cos \theta). In a \(\frac{3\pi}{2}\) form, \(\sin \theta\) changes to its cofunction.
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What is (\cos\left\(\frac{3\pi}{2}+\theta\right\)) equal to?
D. \(\sin \theta\)
(\cos\left\(\frac{3\pi}{2}+\theta\right\)=\sin \theta) होता है। चौथे चतुर्थांश में \(\cos \theta\) धनात्मक माना जाता है। / (\cos\left\(\frac{3\pi}{2}+\theta\right\)=\sin \theta). In the fourth quadrant, \(\cos \theta\) is taken as positive.
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What is (\tan\left\(\frac{3\pi}{2}+\theta\right\)) equal to?
A. -\(\cot \theta\)
\(\frac{3\pi}{2}\) के साथ \(\tan \theta\) \(\cot \theta\) में बदलता है और चौथे चतुर्थांश में चिह्न ऋणात्मक होता है। इसलिए उत्तर \(-\cot \theta\) है। / With \(\frac{3\pi}{2}\), \(\tan \theta\) changes to \(\cot \theta\), and the sign is negative in the fourth quadrant. Therefore the answer is \(-\cot \theta\).
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What is (\sin\(\theta+2\pi\)) equal to?
B. \(\sin \theta\)
\(\sin \theta\) का काल \(2\pi\) है इसलिए \(2\pi\) जोड़ने से मान नहीं बदलता। काल का उपयोग करके बड़े कोण सरल करें। / The period of \(\sin \theta\) is \(2\pi\), so adding \(2\pi\) does not change its value. Use periods to simplify large angles.
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What is (\cos\(\theta+2\pi\)) equal to?
C. \(\cos \theta\)
\(\cos \theta\) का काल \(2\pi\) होता है। इसलिए (\cos\(\theta+2\pi\)=\cos \theta) है। / The period of \(\cos \theta\) is \(2\pi\). Therefore (\cos\(\theta+2\pi\)=\cos \theta).
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What is (\tan\(\theta+\pi\)) equal to?
D. \(\tan \theta\)
\(\tan \theta\) का काल \(\pi\) है इसलिए (\tan\(\theta+\pi\)=\tan \theta)। \(\tan \theta\) के लिए \(2\pi\) की जरूरत नहीं होती। / The period of \(\tan \theta\) is \(\pi\), so (\tan\(\theta+\pi\)=\tan \theta). For \(\tan \theta\), \(2\pi\) is not needed.
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What is (\cot\(\theta+\pi\)) equal to?
A. \(\cot \theta\)
\(\cot \theta\) का काल \(\pi\) होता है। इसलिए (\cot\(\theta+\pi\)) का मान \(\cot \theta\) ही रहता है। / The period of \(\cot \theta\) is \(\pi\). Therefore (\cot\(\theta+\pi\)) remains \(\cot \theta\).
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What is (\sec\(\theta+2\pi\)) equal to?
B. \(\sec \theta\)
\(\sec \theta\) का काल \(2\pi\) है क्योंकि यह \(\cos \theta\) का व्युत्क्रम है। इसलिए \(2\pi\) जोड़ने पर मान वही रहता है। / The period of \(\sec \theta\) is \(2\pi\) because it is the reciprocal of \(\cos \theta\). Hence adding \(2\pi\) keeps the value same.
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What is (\cosec\(\theta+2\pi\)) equal to?
C. \(\cosec \theta\)
\(\cosec \theta\) का काल \(2\pi\) है क्योंकि यह \(\sin \theta\) का व्युत्क्रम है। काल से उत्तर तुरंत मिलता है। / The period of \(\cosec \theta\) is \(2\pi\) because it is the reciprocal of \(\sin \theta\). The period gives the answer directly.
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What is (\sin\(\theta+\pi\)) equal to?
D. -\(\sin \theta\)
(\sin\(\theta+\pi\)=-\sin \theta) होता है। \(\pi\) जोड़ने पर \(\sin \theta\) का चिह्न बदल जाता है। / (\sin\(\theta+\pi\)=-\sin \theta). Adding \(\pi\) changes the sign of \(\sin \theta\).
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What is (\cos\(\theta+\pi\)) equal to?
A. -\(\cos \theta\)
(\cos\(\theta+\pi\)=-\cos \theta) होता है। आधा चक्र जोड़ने से \(\cos \theta\) का चिह्न बदलता है। / (\cos\(\theta+\pi\)=-\cos \theta). Adding half a cycle changes the sign of \(\cos \theta\).
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What is (\sin\(\theta-\pi\)) equal to?
B. -\(\sin \theta\)
(\sin\(\theta-\pi\)=-\sin \theta) होता है। \(\pi\) घटाने पर भी \(\sin \theta\) का चिह्न बदलता है। / (\sin\(\theta-\pi\)=-\sin \theta). Subtracting \(\pi\) also changes the sign of \(\sin \theta\).
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What is (\cos\(\theta-\pi\)) equal to?
C. -\(\cos \theta\)
(\cos\(\theta-\pi\)=-\cos \theta) होता है। \(\pi\) का अंतर \(\cos \theta\) का चिह्न बदल देता है। / (\cos\(\theta-\pi\)=-\cos \theta). A difference of \(\pi\) changes the sign of \(\cos \theta\).
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What is (\tan\(\theta-\pi\)) equal to?
D. \(\tan \theta\)
\(\tan \theta\) का काल \(\pi\) है इसलिए (\tan\(\theta-\pi\)=\tan \theta)। काल घटाने पर भी मान वही रहता है। / The period of \(\tan \theta\) is \(\pi\), so (\tan\(\theta-\pi\)=\tan \theta). Subtracting the period keeps the value same.
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What is (\sin\(2\pi-\theta\)) equal to?
A. -\(\sin \theta\)
(\sin\(2\pi-\theta\)=-\sin \theta) होता है। चौथे चतुर्थांश में \(\sin \theta\) ऋणात्मक होता है। / (\sin\(2\pi-\theta\)=-\sin \theta). In the fourth quadrant, \(\sin \theta\) is negative.
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What is (\cos\(2\pi-\theta\)) equal to?
B. \(\cos \theta\)
(\cos\(2\pi-\theta\)=\cos \theta) होता है। चौथे चतुर्थांश में \(\cos \theta\) धनात्मक होता है। / (\cos\(2\pi-\theta\)=\cos \theta). In the fourth quadrant, \(\cos \theta\) is positive.
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What is (\tan\(2\pi-\theta\)) equal to?
C. -\(\tan \theta\)
(\tan\(2\pi-\theta\)=-\tan \theta) होता है। चौथे चतुर्थांश में \(\tan \theta\) ऋणात्मक होता है। / (\tan\(2\pi-\theta\)=-\tan \theta). In the fourth quadrant, \(\tan \theta\) is negative.
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What is (\sec\(2\pi-\theta\)) equal to?
D. \(\sec \theta\)
\(\sec \theta\) \(\cos \theta\) का व्युत्क्रम है और चौथे चतुर्थांश में धनात्मक रहता है। इसलिए (\sec\(2\pi-\theta\)=\sec \theta) है। / \(\sec \theta\) is the reciprocal of \(\cos \theta\) and remains positive in the fourth quadrant. So (\sec\(2\pi-\theta\)=\sec \theta).
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What is (\cosec\(2\pi-\theta\)) equal to?
A. -\(\cosec \theta\)
\(\cosec \theta\) \(\sin \theta\) का व्युत्क्रम है और चौथे चतुर्थांश में ऋणात्मक होता है। इसलिए मान \(-\cosec \theta\) है। / \(\cosec \theta\) is the reciprocal of \(\sin \theta\) and is negative in the fourth quadrant. Therefore the value is \(-\cosec \theta\).
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What is the maximum value of \(\sin^2 \theta\)?
B. (1)
\(\sin \theta\) का मान ([-1,1]) में होता है इसलिए \(\sin^2 \theta\) का अधिकतम मान (1) है। वर्ग करने पर मान ऋणात्मक नहीं रहता। / The value of \(\sin \theta\) lies in ([-1,1]), so the maximum value of \(\sin^2 \theta\) is (1). Squaring makes the value non-negative.
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What is the minimum value of \(\cos^2 \theta\)?
C. (0)
\(\cos^2 \theta\) हमेशा (0) या धनात्मक होता है। इसका न्यूनतम मान (0) है। / \(\cos^2 \theta\) is always (0) or positive. Its minimum value is (0).
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What type of value does \(\sin^2 \theta+\cos^2 \theta\) have?
D. स्थिरConstant
\(\sin^2 \theta+\cos^2 \theta=1\) होता है इसलिए यह स्थिर मान है। मूल सर्वसमिका सीधे पहचानें। / \(\sin^2 \theta+\cos^2 \theta=1\), so it is a constant value. Identify the basic identity directly.
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If \(\sin \theta=0\), what is the value of \(\cos^2 \theta\)?
A. (1)
\(\sin^2 \theta+\cos^2 \theta=1\) में \(\sin \theta=0\) रखने पर \(\cos^2 \theta=1\) मिलता है। ऐसे प्रश्नों में पहचान सूत्र लगाएँ। / Putting \(\sin \theta=0\) in \(\sin^2 \theta+\cos^2 \theta=1\) gives \(\cos^2 \theta=1\). Use the identity in such questions.
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If \(\cos \theta=0\), what is the value of \(\sin^2 \theta\)?
B. (1)
\(\sin^2 \theta+\cos^2 \theta=1\) में \(\cos \theta=0\) रखने पर \(\sin^2 \theta=1\) आता है। ज्ञात मान को सूत्र में रखें। / Putting \(\cos \theta=0\) in \(\sin^2 \theta+\cos^2 \theta=1\) gives \(\sin^2 \theta=1\). Substitute the known value in the formula.
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What is \(\frac{1}{1+\tan^2 \theta}\) equal to?
C. \(\cos^2 \theta\)
\(\1+\tan^2 \theta=\sec^2 \theta\) और \(\frac{1}{\sec^2 \theta}=\cos^2 \theta\) होता है। व्युत्क्रम लेते समय सावधानी रखें। / \(\1+\tan^2 \theta=\sec^2 \theta\) and \(\frac{1}{\sec^2 \theta}=\cos^2 \theta\). Be careful while taking reciprocals.
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What is \(\frac{1}{1+\cot^2 \theta}\) equal to?
D. \(\sin^2 \theta\)
\(\1+\cot^2 \theta=\cosec^2 \theta\) और \(\frac{1}{\cosec^2 \theta}=\sin^2 \theta\) होता है। पहचान सूत्र से भिन्न सरल करें। / \(\1+\cot^2 \theta=\cosec^2 \theta\) and \(\frac{1}{\cosec^2 \theta}=\sin^2 \theta\). Simplify fractions using identities.
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What is \(\sec^2 \theta-1\) equal to?
A. \(\tan^2 \theta\)
\(\sec^2 \theta=1+\tan^2 \theta\) से \(\sec^2 \theta-1=\tan^2 \theta\) मिलता है। सूत्र को सरल रूप में बदलें। / From \(\sec^2 \theta=1+\tan^2 \theta\), we get \(\sec^2 \theta-1=\tan^2 \theta\). Rearrange the formula into a simple form.
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What is \(\cosec^2 \theta-1\) equal to?
B. \(\cot^2 \theta\)
\(\cosec^2 \theta=1+\cot^2 \theta\) से \(\cosec^2 \theta-1=\cot^2 \theta\) होता है। समान पहचान सूत्रों की जोड़ी बनाकर अभ्यास करें। / From \(\cosec^2 \theta=1+\cot^2 \theta\), \(\cosec^2 \theta-1=\cot^2 \theta\). Practice similar identities in pairs.
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What is \(\frac{\sin \theta}{\cosec \theta}\) equal to?
C. \(\sin^2 \theta\)
\(\cosec \theta=\frac{1}{\sin \theta}\) इसलिए \(\frac{\sin \theta}{\cosec \theta}=\sin^2 \theta\) है। व्युत्क्रम संबंध को भिन्न में बदलें। / Since \(\cosec \theta=\frac{1}{\sin \theta}\), \(\frac{\sin \theta}{\cosec \theta}=\sin^2 \theta\). Convert reciprocal relations into fractions.
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What is \(\frac{\cos \theta}{\sec \theta}\) equal to?
D. \(\cos^2 \theta\)
\(\sec \theta=\frac{1}{\cos \theta}\) इसलिए \(\frac{\cos \theta}{\sec \theta}=\cos^2 \theta\) होता है। व्युत्क्रम लिखने से सरलीकरण आसान होता है। / Since \(\sec \theta=\frac{1}{\cos \theta}\), \(\frac{\cos \theta}{\sec \theta}=\cos^2 \theta\). Writing the reciprocal makes simplification easy.
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What is the value of \(\sin \theta \cdot \cosec \theta\)?
A. (1)
\(\sin \theta\) और \(\cosec \theta\) परस्पर व्युत्क्रम हैं इसलिए उनका गुणनफल (1) होता है। व्युत्क्रम जोड़ों को याद रखें। / \(\sin \theta\) and \(\cosec \theta\) are reciprocals, so their product is (1). Remember reciprocal pairs.
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What is the value of \(\cos \theta \cdot \sec \theta\)?
B. (1)
\(\cos \theta\) और \(\sec \theta\) एक-दूसरे के व्युत्क्रम हैं। इसलिए उनका गुणनफल (1) है। / \(\cos \theta\) and \(\sec \theta\) are reciprocals of each other. Therefore their product is (1).
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What is the value of \(\tan \theta \cdot \cot \theta\)?
C. (1)
\(\tan \theta\) और \(\cot \theta\) परस्पर व्युत्क्रम हैं। इसलिए गुणनफल (1) होता है। / \(\tan \theta\) and \(\cot \theta\) are reciprocals. Hence their product is (1).
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What is \(\frac{\tan \theta}{\sec \theta}\) equal to?
D. \(\sin \theta\)
\(\tan \theta=\frac{\sin \theta}{\cos \theta}\) और \(\sec \theta=\frac{1}{\cos \theta}\) रखने पर उत्तर \(\sin \theta\) मिलता है। भाग को भिन्न में बदलकर काटें। / Using \(\tan \theta=\frac{\sin \theta}{\cos \theta}\) and \(\sec \theta=\frac{1}{\cos \theta}\), the answer is \(\sin \theta\). Convert division into fractions and cancel.
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What is \(\frac{\cot \theta}{\cosec \theta}\) equal to?
A. \(\cos \theta\)
\(\cot \theta=\frac{\cos \theta}{\sin \theta}\) और \(\cosec \theta=\frac{1}{\sin \theta}\) रखने पर \(\cos \theta\) मिलता है। मानक रूप लिखना सबसे सुरक्षित तरीका है। / Using \(\cot \theta=\frac{\cos \theta}{\sin \theta}\) and \(\cosec \theta=\frac{1}{\sin \theta}\), we get \(\cos \theta\). Writing standard forms is the safest method.
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What is \(\frac{\sec \theta}{\tan \theta}\) equal to?
B. \(\cosec \theta\)
\(\sec \theta=\frac{1}{\cos \theta}\) और \(\tan \theta=\frac{\sin \theta}{\cos \theta}\) से \(\frac{1}{\sin \theta}=\cosec \theta\) मिलता है। भिन्नों को सावधानी से सरल करें। / From \(\sec \theta=\frac{1}{\cos \theta}\) and \(\tan \theta=\frac{\sin \theta}{\cos \theta}\), we get \(\frac{1}{\sin \theta}=\cosec \theta\). Simplify fractions carefully.
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What is \(\frac{\cosec \theta}{\cot \theta}\) equal to?
A. \(\sec \theta\)
\(\cosec \theta=\frac{1}{\sin \theta}\) और \(\cot \theta=\frac{\cos \theta}{\sin \theta}\) से \(\frac{1}{\cos \theta}=\sec \theta\) मिलता है। अंतिम उत्तर को मानक फलन में लिखें। / From \(\cosec \theta=\frac{1}{\sin \theta}\) and \(\cot \theta=\frac{\cos \theta}{\sin \theta}\), we get \(\frac{1}{\cos \theta}=\sec \theta\). Write the final answer as a standard function.
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What is the period of the function \(\sin x\)?
C. \(2\pi\)
\(\sin x\) हर \(2\pi\) के बाद अपना मान दोहराता है। परीक्षा में काल पूछने पर सबसे पहले मूल फलन पहचानें। / \(\sin x\) repeats its value after every \(2\pi\). In exams first identify the basic function.
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What is the range of the function \(\cos x\)?
B. ([-1,1])
\(\cos x\) का मान हमेशा (-1) और (1) के बीच रहता है। परिसर में अंतिम मान भी शामिल होते हैं। / The value of \(\cos x\) always lies between (-1) and (1). The end values are included in the range.
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Which of the following functions is odd?
C. \(\sin x\)
क्योंकि (\sin(-x)=-\sin x), इसलिए \(\sin x\) विषम फलन है। विषम फलन में (f(-x)=-f(x)) होता है। / Since (\sin(-x)=-\sin x), \(\sin x\) is an odd function. For an odd function, (f(-x)=-f(x)).
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Which of the following functions is even?
D. \(\cos x\)
क्योंकि (\cos(-x)=\cos x), इसलिए \(\cos x\) सम फलन है। सम फलन में (f(-x)=f(x)) होता है। / Since (\cos(-x)=\cos x), \(\cos x\) is an even function. For an even function, (f(-x)=f(x)).
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What is the value of the identity \(\sin^2 x+\cos^2 x\)?
B. (1)
मूल पहचान \(\sin^2 x+\cos^2 x=1\) होती है। इसे याद रखना कई प्रश्नों में काम आता है। / The basic identity is \(\sin^2 x+\cos^2 x=1\). Remembering it helps in many questions.
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What is the period of the function \(\tan x\)?
A. \(\pi\)
\(\tan x\) हर \(\pi\) के बाद अपना मान दोहराता है। \(\tan x\) और \(\cot x\) का काल \(\pi\) होता है। / \(\tan x\) repeats its value after every \(\pi\). The period of \(\tan x\) and \(\cot x\) is \(\pi\).
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The function \(\sec x\) is the reciprocal of which function?
C. \(\cos x\)
\(\sec x=\frac{1}{\cos x}\) होता है। व्युत्क्रम संबंधों को जोड़ी बनाकर याद करें। / \(\sec x=\frac{1}{\cos x}\). Learn reciprocal relations in pairs.
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The function \(\cosec x\) is the reciprocal of which function?
B. \(\sin x\)
\(\cosec x=\frac{1}{\sin x}\) होता है। ध्यान रखें कि \(\cosec x\) का संबंध \(\sin x\) से है। / \(\cosec x=\frac{1}{\sin x}\). Remember that \(\cosec x\) is related to \(\sin x\).
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How is \(\tan x\) written in terms of \(\sin x\) and \(\cos x\)?
C. \(\frac{\sin x}{\cos x}\)
\(\tan x=\frac{\sin x}{\cos x}\) मूल अनुपात पहचान है। ऐसे प्रश्नों में अंश और हर को ध्यान से देखें। / \(\tan x=\frac{\sin x}{\cos x}\) is a basic quotient identity. In such questions, check numerator and denominator carefully.
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How is \(\cot x\) written in terms of \(\sin x\) and \(\cos x\)?
B. \(\frac{\cos x}{\sin x}\)
\(\cot x=\frac{\cos x}{\sin x}\) होता है। \(\tan x\) और \(\cot x\) एक-दूसरे के व्युत्क्रम हैं। / \(\cot x=\frac{\cos x}{\sin x}\). \(\tan x\) and \(\cot x\) are reciprocals of each other.
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What is the range of the function \(\sin x\)?
A. ([-1,1])
\(\sin x\) का मान (-1) से (1) तक होता है। ग्राफ याद रखने से परिसर जल्दी मिलता है। / The value of \(\sin x\) lies from (-1) to (1). Remembering the graph helps find the range quickly.
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What is the range of the function \(\tan x\)?
C. (\(-\infty,\infty\))
\(\tan x\) सभी वास्तविक मान ले सकता है। इसलिए इसका परिसर (\(-\infty,\infty\)) है। / \(\tan x\) can take all real values. Therefore, its range is (\(-\infty,\infty\)).
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What is the value of \(\sec^2 x-\tan^2 x\)?
D. (1)
क्योंकि \(\sec^2 x=1+\tan^2 x\), इसलिए \(\sec^2 x-\tan^2 x=1\)। पहचान को सही दिशा में प्रयोग करें। / Since \(\sec^2 x=1+\tan^2 x\), \(\sec^2 x-\tan^2 x=1\). Use the identity in the correct direction.
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What is the value of \(\cosec^2 x-\cot^2 x\)?
B. (1)
क्योंकि \(\cosec^2 x=1+\cot^2 x\), अंतर (1) होगा। यह पहचान \(\sin^2 x+\cos^2 x=1\) से जुड़ी है। / Since \(\cosec^2 x=1+\cot^2 x\), the difference is (1). This identity is linked to \(\sin^2 x+\cos^2 x=1\).
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If \(\sin x=0\), what is the value of \(\cos x\) at (x=0)?
C. (1)
(x=0) पर \(\cos 0=1\) होता है। मानक कोणों के मान याद रखना आसान प्रश्नों में मदद करता है। / At (x=0), \(\cos 0=1\). Remembering standard angle values helps in easy questions.
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What is the value of \(\sin 0\)?
B. (0)
\(\sin 0=0\) एक मानक मान है। परीक्षा में (0), \(\frac{\pi}{2}\) और \(\pi\) के मान जरूर याद रखें। / \(\sin 0=0\) is a standard value. In exams, remember the values for (0), \(\frac{\pi}{2}\), and \(\pi\).
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What is the value of \(\tan 0\)?
C. (0)
\(\tan 0=\frac{\sin 0}{\cos 0}=\frac{0}{1}=0\)। अनुपात पहचान से मान आसानी से निकलेगा। / \(\tan 0=\frac{\sin 0}{\cos 0}=\frac{0}{1}=0\). The quotient identity gives the value easily.
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What is the value of \(\sin \frac{\pi}{2}\)?
B. (1)
\(\frac{\pi}{2}\) पर \(\sin x\) का मान (1) होता है। यूनिट सर्कल से यह मान जल्दी याद होता है। / At \(\frac{\pi}{2}\), the value of \(\sin x\) is (1). The unit circle helps remember this value quickly.
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What is the value of \(\cos \frac{\pi}{2}\)?
C. (0)
\(\frac{\pi}{2}\) पर \(\cos x\) का मान (0) है। मानक कोणों की तालिका से गलती कम होती है। / At \(\frac{\pi}{2}\), the value of \(\cos x\) is (0). A standard angle table reduces mistakes.
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What is (\sin\(\pi-x\)) equal to?
C. \(\sin x\)
\(\pi-x\) दूसरे चतुर्थांश में होता है और वहाँ \(\sin x\) धनात्मक रहता है। इसलिए (\sin\(\pi-x\)=\sin x)। / \(\pi-x\) lies in the second quadrant and \(\sin x\) remains positive there. Hence (\sin\(\pi-x\)=\sin x).
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What is (\cos\(\pi-x\)) equal to?
B. \(-\cos x\)
दूसरे चतुर्थांश में \(\cos x\) ऋणात्मक होता है। इसलिए (\cos\(\pi-x\)=-\cos x)। / In the second quadrant, \(\cos x\) is negative. Therefore, (\cos\(\pi-x\)=-\cos x).
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What is (\tan\(\pi+x\)) equal to?
C. \(\tan x\)
\(\tan x\) का काल \(\pi\) है, इसलिए (\tan\(\pi+x\)=\tan x)। काल का उपयोग करके उत्तर जल्दी मिलता है। / The period of \(\tan x\) is \(\pi\), so (\tan\(\pi+x\)=\tan x). Using period gives the answer quickly.
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B. \(-\sin x\)
\(\sin x\) विषम फलन है, इसलिए (\sin(-x)=-\sin x)। सम-विषम गुण जल्दी पहचानें। / \(\sin x\) is an odd function, so (\sin(-x)=-\sin x). Identify even-odd properties quickly.
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C. \(\cos x\)
\(\cos x\) सम फलन है, इसलिए (\cos(-x)=\cos x)। ऋण कोण वाले प्रश्नों में सम-विषम गुण लगाएँ। / \(\cos x\) is an even function, so (\cos(-x)=\cos x). Use even-odd properties for negative angle questions.
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B. \(-\tan x\)
\(\tan x\) विषम फलन है, इसलिए (\tan(-x)=-\tan x)। इसे \(\frac{\sin x}{\cos x}\) से भी समझ सकते हैं। / \(\tan x\) is an odd function, so (\tan(-x)=-\tan x). This can also be understood from \(\frac{\sin x}{\cos x}\).
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What is the sign of \(\sin x\) in the first quadrant?
C. धनात्मकPositive
पहले चतुर्थांश में सभी त्रिकोणमितीय फलन धनात्मक होते हैं। चतुर्थांश संकेत तालिका याद रखें। / In the first quadrant, all trigonometric functions are positive. Remember the quadrant sign table.
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What is the sign of \(\cos x\) in the second quadrant?
B. ऋणात्मकNegative
दूसरे चतुर्थांश में केवल \(\sin x\) और \(\cosec x\) धनात्मक होते हैं। इसलिए \(\cos x\) ऋणात्मक होता है। / In the second quadrant, only \(\sin x\) and \(\cosec x\) are positive. Therefore, \(\cos x\) is negative.
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What is the sign of \(\tan x\) in the third quadrant?
C. धनात्मकPositive
तीसरे चतुर्थांश में \(\tan x\) और \(\cot x\) धनात्मक होते हैं। संकेत नियम से उत्तर सीधे मिलता है। / In the third quadrant, \(\tan x\) and \(\cot x\) are positive. The sign rule gives the answer directly.
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What is the sign of \(\sin x\) in the fourth quadrant?
B. ऋणात्मकNegative
चौथे चतुर्थांश में \(\cos x\) और \(\sec x\) धनात्मक होते हैं। इसलिए \(\sin x\) ऋणात्मक होता है। / In the fourth quadrant, \(\cos x\) and \(\sec x\) are positive. Hence \(\sin x\) is negative.
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What is (\sin\(2\pi+x\)) equal to?
A. \(\sin x\)
\(\sin x\) का काल \(2\pi\) है, इसलिए (\sin\(2\pi+x\)=\sin x)। काल वाली पहचान याद रखें। / The period of \(\sin x\) is \(2\pi\), so (\sin\(2\pi+x\)=\sin x). Remember periodic identities.
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What is (\cos\(2\pi+x\)) equal to?
C. \(\cos x\)
\(\cos x\) हर \(2\pi\) पर दोहरता है। इसलिए (\cos\(2\pi+x\)=\cos x)। / \(\cos x\) repeats after every \(2\pi\). Therefore, (\cos\(2\pi+x\)=\cos x).
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What is (\sin\(\frac{\pi}{2}-x\)) equal to?
C. \(\cos x\)
पूरक कोण पहचान से (\sin\(\frac{\pi}{2}-x\)=\cos x) होता है। \(\frac{\pi}{2}\) के साथ फलन बदलता है। / By the complementary angle identity, (\sin\(\frac{\pi}{2}-x\)=\cos x). With \(\frac{\pi}{2}\), the function changes.
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What is (\cos\(\frac{\pi}{2}-x\)) equal to?
D. \(\sin x\)
पूरक कोण पहचान से (\cos\(\frac{\pi}{2}-x\)=\sin x) होता है। ऐसे प्रश्नों में \(\sin x\) और \(\cos x\) आपस में बदलते हैं। / By the complementary angle identity, (\cos\(\frac{\pi}{2}-x\)=\sin x). In such questions, \(\sin x\) and \(\cos x\) interchange.
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What is (\tan\(\frac{\pi}{2}-x\)) equal to?
B. \(\cot x\)
\(\frac{\pi}{2}\) के पूरक कोण में \(\tan x\) बदलकर \(\cot x\) हो जाता है। पूरक पहचान याद रखें। / For a complementary angle with \(\frac{\pi}{2}\), \(\tan x\) changes to \(\cot x\). Remember cofunction identities.
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What is the period of the function \(\sec x\)?
C. \(2\pi\)
\(\sec x\) का काल \(\cos x\) जैसा \(2\pi\) होता है। व्युत्क्रम फलन का काल मूल फलन से जुड़ा रहता है। / The period of \(\sec x\) is \(2\pi\), like \(\cos x\). The reciprocal function keeps the related basic period.
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What is the period of the function \(\cot x\)?
A. \(\pi\)
\(\cot x\) हर \(\pi\) के बाद अपना मान दोहराता है। \(\tan x\) और \(\cot x\) दोनों का काल \(\pi\) है। / \(\cot x\) repeats its value after every \(\pi\). Both \(\tan x\) and \(\cot x\) have period \(\pi\).
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What is the period of the function \(\cosec x\)?
C. \(2\pi\)
\(\cosec x\) \(\sin x\) का व्युत्क्रम है और इसका काल \(2\pi\) है। व्युत्क्रम संबंध से काल पहचानें। / \(\cosec x\) is the reciprocal of \(\sin x\) and its period is \(2\pi\). Use reciprocal relation to identify the period.
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What is the range of \(\sec x\)?
D. (\(-\infty,-1]\cup[1,\infty\))
\(\sec x=\frac{1}{\cos x}\) है और \(\cos x\) का मान ([-1,1]) में होता है। इसलिए \(\sec x\) का मान (1) से बड़ा या (-1) से छोटा होता है। / \(\sec x=\frac{1}{\cos x}\) and \(\cos x\) lies in ([-1,1]). Hence \(\sec x\) is greater than or equal to (1) or less than or equal to (-1).
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What is the range of \(\cosec x\)?
A. (\(-\infty,-1]\cup[1,\infty\))
\(\cosec x=\frac{1}{\sin x}\) है। इसलिए इसका परिसर (\(-\infty,-1]\cup[1,\infty\)) होता है। / \(\cosec x=\frac{1}{\sin x}\). Therefore, its range is (\(-\infty,-1]\cup[1,\infty\)).
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Where is \(\tan x\) undefined?
A. \(\cos x=0\)
\(\tan x=\frac{\sin x}{\cos x}\) है, इसलिए \(\cos x=0\) पर यह अपरिभाषित होता है। हर शून्य होने पर भिन्न परिभाषित नहीं रहती। / \(\tan x=\frac{\sin x}{\cos x}\), so it is undefined when \(\cos x=0\). A fraction is not defined when the denominator is zero.
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Where is \(\cot x\) undefined?
B. \(\sin x=0\)
\(\cot x=\frac{\cos x}{\sin x}\) है, इसलिए \(\sin x=0\) पर यह अपरिभाषित होता है। हर को हमेशा ध्यान से देखें। / \(\cot x=\frac{\cos x}{\sin x}\), so it is undefined when \(\sin x=0\). Always check the denominator carefully.
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Where is \(\sec x\) undefined?
C. \(\cos x=0\)
\(\sec x=\frac{1}{\cos x}\) है, इसलिए \(\cos x=0\) पर यह अपरिभाषित होता है। व्युत्क्रम फलनों में हर शून्य नहीं हो सकता। / \(\sec x=\frac{1}{\cos x}\), so it is undefined when \(\cos x=0\). In reciprocal functions, the denominator cannot be zero.
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Where is \(\cosec x\) undefined?
B. \(\sin x=0\)
\(\cosec x=\frac{1}{\sin x}\) है, इसलिए \(\sin x=0\) पर यह अपरिभाषित होता है। व्युत्क्रम में नीचे वाला फलन शून्य नहीं होना चाहिए। / \(\cosec x=\frac{1}{\sin x}\), so it is undefined when \(\sin x=0\). In a reciprocal, the denominator function must not be zero.
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If \(\sin x=\frac{3}{5}\) and (x) is in the first quadrant, what is the value of \(\cos x\)?
C. \(\frac{4}{5}\)
\(\sin^2 x+\cos^2 x=1\) से \(\cos x=\frac{4}{5}\) मिलता है। प्रथम चतुर्थांश में \(\cos x\) धनात्मक होता है। / Using \(\sin^2 x+\cos^2 x=1\), \(\cos x=\frac{4}{5}\). In the first quadrant, \(\cos x\) is positive.
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If \(\cos x=\frac{12}{13}\) and (x) is in the first quadrant, what is the value of \(\sin x\)?
A. \(\frac{5}{13}\)
\(\sin^2 x=1-\cos^2 x\) से \(\sin x=\frac{5}{13}\) मिलता है। प्रथम चतुर्थांश में मान धनात्मक लिया जाता है। / From \(\sin^2 x=1-\cos^2 x\), \(\sin x=\frac{5}{13}\). In the first quadrant, the positive value is taken.
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If \(\tan x=\frac{7}{24}\) and (x) is in the first quadrant, what is the value of \(\sec x\)?
B. \(\frac{25}{24}\)
\(\sec^2 x=1+\tan^2 x\) लगाने पर \(\sec x=\frac{25}{24}\) मिलता है। प्रथम चतुर्थांश में \(\sec x\) धनात्मक होता है। / Using \(\sec^2 x=1+\tan^2 x\), \(\sec x=\frac{25}{24}\). In the first quadrant, \(\sec x\) is positive.
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If \(\cot x=\frac{8}{15}\) and (x) is in the first quadrant, what is the value of \(\cosec x\)?
B. \(\frac{17}{15}\)
\(\cosec^2 x=1+\cot^2 x\) से \(\cosec x=\frac{17}{15}\) मिलता है। प्रथम चतुर्थांश में धनात्मक मान लें। / From \(\cosec^2 x=1+\cot^2 x\), \(\cosec x=\frac{17}{15}\). Take the positive value in the first quadrant.
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What is (\sin\(\pi+x\)) equal to?
C. \(-\sin x\)
\(\pi+x\) तीसरे चतुर्थांश में आता है और वहाँ \(\sin x\) ऋणात्मक होता है। इसलिए (\sin\(\pi+x\)=-\sin x)। / \(\pi+x\) lies in the third quadrant and \(\sin x\) is negative there. Hence (\sin\(\pi+x\)=-\sin x).
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What is (\cos\(\pi+x\)) equal to?
B. \(-\cos x\)
\(\pi+x\) तीसरे चतुर्थांश में है और वहाँ \(\cos x\) ऋणात्मक होता है। इसलिए (\cos\(\pi+x\)=-\cos x)। / \(\pi+x\) is in the third quadrant and \(\cos x\) is negative there. Therefore, (\cos\(\pi+x\)=-\cos x).
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Which of the following identities is correct?
D. \(1+\tan^2 x=\sec^2 x\)
सही पहचान \(1+\tan^2 x=\sec^2 x\) है। पहचान आधारित प्रश्नों में चिन्ह और भिन्न को सावधानी से देखें। / The correct identity is \(1+\tan^2 x=\sec^2 x\). In identity-based questions, check signs and fractions carefully.
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