In the second quadrant, \(\sin \theta\) and \(\cosec \theta\) are positive. Knowing sign rules helps solve such questions quickly.
Step 2
Why this answer is correct
The correct answer is B. \(\sin \theta\). In the second quadrant, \(\sin \theta\) and \(\cosec \theta\) are positive. Knowing sign rules helps solve such questions quickly.
Step 3
Exam Tip
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 third quadrant, \(\tan \theta\) and \(\cot \theta\) are positive. The sign changes according to the quadrant.
Step 2
Why this answer is correct
The correct answer is C. धनात्मक / Positive. In the third quadrant, \(\tan \theta\) and \(\cot \theta\) are positive. The sign changes according to the quadrant.
Step 3
Exam Tip
तीसरे चतुर्थांश में \(\tan \theta\) और \(\cot \theta\) धनात्मक होते हैं। चतुर्थांश के अनुसार चिह्न बदलता है।
In the fourth quadrant, \(\cos \theta\) and \(\sec \theta\) are positive. Remember sign rules using a small table.
Step 2
Why this answer is correct
The correct answer is D. \(\cos \theta\). In the fourth quadrant, \(\cos \theta\) and \(\sec \theta\) are positive. Remember sign rules using a small table.
Step 3
Exam Tip
चौथे चतुर्थांश में \(\cos \theta\) और \(\sec \theta\) धनात्मक होते हैं। संकेत नियम को छोटी तालिका से याद करें।
With \(\frac{\pi}{2}\), \(\sin \theta\) changes to its cofunction \(\cos \theta\). The sign remains positive in the second quadrant.
Step 2
Why this answer is correct
The correct answer is A. \(\cos \theta\). With \(\frac{\pi}{2}\), \(\sin \theta\) changes to its cofunction \(\cos \theta\). The sign remains positive in the second quadrant.
Step 3
Exam Tip
\(\frac{\pi}{2}\) के साथ \(\sin \theta\) सह-फलन \(\cos \theta\) में बदलता है। दूसरे चतुर्थांश में चिह्न धनात्मक रहता 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.
Step 2
Why this answer is correct
The correct answer is B. -\(\sin \theta\). (\cos\left\(\frac{\pi}{2}+\theta\right\)=-\sin \theta). At \(\frac{\pi}{2}\), the function changes and the sign is decided by the quadrant.
Step 3
Exam Tip
(\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।
The cofunction of \(\tan \theta\) is \(\cot \theta\), and the sign is negative in the second quadrant. Therefore the answer is \(-\cot \theta\).
Step 2
Why this answer is correct
The correct answer is C. -\(\cot \theta\). The cofunction of \(\tan \theta\) is \(\cot \theta\), and the sign is negative in the second quadrant. Therefore the answer is \(-\cot \theta\).
Step 3
Exam Tip
\(\tan \theta\) का सह-फलन \(\cot \theta\) है और दूसरे चतुर्थांश में चिह्न ऋणात्मक होता है। इसलिए उत्तर \(-\cot \theta\) है।
The cofunction of \(\cot \theta\) is \(\tan \theta\), and the sign is negative in the second quadrant. In such questions, first identify the cofunction.
Step 2
Why this answer is correct
The correct answer is D. -\(\tan \theta\). The cofunction of \(\cot \theta\) is \(\tan \theta\), and the sign is negative in the second quadrant. In such questions, first identify the cofunction.
Step 3
Exam Tip
\(\cot \theta\) का सह-फलन \(\tan \theta\) है और दूसरे चतुर्थांश में चिह्न ऋणात्मक होता है। ऐसे प्रश्नों में पहले सह-फलन पहचानें।
The cofunction of \(\sec \theta\) is \(\cosec \theta\), and \(\sec \theta\) is negative in the second quadrant. Hence the value is \(-\cosec \theta\).
Step 2
Why this answer is correct
The correct answer is A. -\(\cosec \theta\). The cofunction of \(\sec \theta\) is \(\cosec \theta\), and \(\sec \theta\) is negative in the second quadrant. Hence the value is \(-\cosec \theta\).
Step 3
Exam Tip
\(\sec \theta\) का सह-फलन \(\cosec \theta\) है और दूसरे चतुर्थांश में \(\sec \theta\) ऋणात्मक होता है। इसलिए मान \(-\cosec \theta\) है।
The cofunction of \(\cosec \theta\) is \(\sec \theta\), and the sign remains positive in the second quadrant. Remember cofunction pairs.
Step 2
Why this answer is correct
The correct answer is B. \(\sec \theta\). The cofunction of \(\cosec \theta\) is \(\sec \theta\), and the sign remains positive in the second quadrant. Remember cofunction pairs.
Step 3
Exam Tip
\(\cosec \theta\) का सह-फलन \(\sec \theta\) है और दूसरे चतुर्थांश में चिह्न धनात्मक रहता है। सह-फलन जोड़ों को याद रखें।
With \(\frac{3\pi}{2}\), \(\sin \theta\) changes to a cofunction and the sign is negative in the third quadrant. So the answer is \(-\cos \theta\).
Step 2
Why this answer is correct
The correct answer is C. -\(\cos \theta\). With \(\frac{3\pi}{2}\), \(\sin \theta\) changes to a cofunction and the sign is negative in the third quadrant. So the answer is \(-\cos \theta\).
Step 3
Exam Tip
\(\frac{3\pi}{2}\) के साथ \(\sin \theta\) सह-फलन में बदलता है और तीसरे चतुर्थांश में चिह्न ऋणात्मक होता है। इसलिए उत्तर \(-\cos \theta\) है।
With \(\frac{3\pi}{2}\), \(\cos \theta\) changes to the cofunction \(\sin \theta\). In the third quadrant, \(\cos \theta\) is negative.
Step 2
Why this answer is correct
The correct answer is A. -\(\sin \theta\). With \(\frac{3\pi}{2}\), \(\cos \theta\) changes to the cofunction \(\sin \theta\). In the third quadrant, \(\cos \theta\) is negative.
Step 3
Exam Tip
\(\frac{3\pi}{2}\) के साथ \(\cos \theta\) सह-फलन \(\sin \theta\) में बदलता है। तीसरे चतुर्थांश में \(\cos \theta\) ऋणात्मक होता है।
With \(\frac{3\pi}{2}\), \(\tan \theta\) changes to \(\cot \theta\). In the third quadrant, \(\tan \theta\) is positive.
Step 2
Why this answer is correct
The correct answer is B. \(\cot \theta\). With \(\frac{3\pi}{2}\), \(\tan \theta\) changes to \(\cot \theta\). In the third quadrant, \(\tan \theta\) is positive.
Step 3
Exam Tip
\(\frac{3\pi}{2}\) के साथ \(\tan \theta\) \(\cot \theta\) में बदलता है। तीसरे चतुर्थांश में \(\tan \theta\) धनात्मक होता है।
(\sin\left\(\frac{3\pi}{2}+\theta\right\)=-\cos \theta). In a \(\frac{3\pi}{2}\) form, \(\sin \theta\) changes to its cofunction.
Step 2
Why this answer is correct
The correct answer is C. -\(\cos \theta\). (\sin\left\(\frac{3\pi}{2}+\theta\right\)=-\cos \theta). In a \(\frac{3\pi}{2}\) form, \(\sin \theta\) changes to its cofunction.
Step 3
Exam Tip
(\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।
(\cos\left\(\frac{3\pi}{2}+\theta\right\)=\sin \theta). In the fourth quadrant, \(\cos \theta\) is taken as positive.
Step 2
Why this answer is correct
The correct answer is D. \(\sin \theta\). (\cos\left\(\frac{3\pi}{2}+\theta\right\)=\sin \theta). In the fourth quadrant, \(\cos \theta\) is taken as positive.
Step 3
Exam Tip
(\cos\left\(\frac{3\pi}{2}+\theta\right\)=\sin \theta) होता है। चौथे चतुर्थांश में \(\cos \theta\) धनात्मक माना जाता 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 \(-\cot \theta\).
Step 2
Why this answer is correct
The correct answer is A. -\(\cot \theta\). With \(\frac{3\pi}{2}\), \(\tan \theta\) changes to \(\cot \theta\), and the sign is negative in the fourth quadrant. Therefore the answer is \(-\cot \theta\).
Step 3
Exam Tip
\(\frac{3\pi}{2}\) के साथ \(\tan \theta\) \(\cot \theta\) में बदलता है और चौथे चतुर्थांश में चिह्न ऋणात्मक होता है। इसलिए उत्तर \(-\cot \theta\) है।
The period of \(\sin \theta\) is \(2\pi\), so adding \(2\pi\) does not change its value. Use periods to simplify large angles.
Step 2
Why this answer is correct
The correct answer is B. \(\sin \theta\). The period of \(\sin \theta\) is \(2\pi\), so adding \(2\pi\) does not change its value. Use periods to simplify large angles.
Step 3
Exam Tip
\(\sin \theta\) का काल \(2\pi\) है इसलिए \(2\pi\) जोड़ने से मान नहीं बदलता। काल का उपयोग करके बड़े कोण सरल करें।
The period of \(\tan \theta\) is \(\pi\), so (\tan\(\theta+\pi\)=\tan \theta). For \(\tan \theta\), \(2\pi\) is not needed.
Step 2
Why this answer is correct
The correct answer is D. \(\tan \theta\). The period of \(\tan \theta\) is \(\pi\), so (\tan\(\theta+\pi\)=\tan \theta). For \(\tan \theta\), \(2\pi\) is not needed.
Step 3
Exam Tip
\(\tan \theta\) का काल \(\pi\) है इसलिए (\tan\(\theta+\pi\)=\tan \theta)। \(\tan \theta\) के लिए \(2\pi\) की जरूरत नहीं होती।
The period of \(\sec \theta\) is \(2\pi\) because it is the reciprocal of \(\cos \theta\). Hence adding \(2\pi\) keeps the value same.
Step 2
Why this answer is correct
The correct answer is B. \(\sec \theta\). The period of \(\sec \theta\) is \(2\pi\) because it is the reciprocal of \(\cos \theta\). Hence adding \(2\pi\) keeps the value same.
Step 3
Exam Tip
\(\sec \theta\) का काल \(2\pi\) है क्योंकि यह \(\cos \theta\) का व्युत्क्रम है। इसलिए \(2\pi\) जोड़ने पर मान वही रहता है।
The period of \(\cosec \theta\) is \(2\pi\) because it is the reciprocal of \(\sin \theta\). The period gives the answer directly.
Step 2
Why this answer is correct
The correct answer is C. \(\cosec \theta\). The period of \(\cosec \theta\) is \(2\pi\) because it is the reciprocal of \(\sin \theta\). The period gives the answer directly.
Step 3
Exam Tip
\(\cosec \theta\) का काल \(2\pi\) है क्योंकि यह \(\sin \theta\) का व्युत्क्रम है। काल से उत्तर तुरंत मिलता है।
The period of \(\tan \theta\) is \(\pi\), so (\tan\(\theta-\pi\)=\tan \theta). Subtracting the period keeps the value same.
Step 2
Why this answer is correct
The correct answer is D. \(\tan \theta\). The period of \(\tan \theta\) is \(\pi\), so (\tan\(\theta-\pi\)=\tan \theta). Subtracting the period keeps the value same.
Step 3
Exam Tip
\(\tan \theta\) का काल \(\pi\) है इसलिए (\tan\(\theta-\pi\)=\tan \theta)। काल घटाने पर भी मान वही रहता है।
\(\sec \theta\) is the reciprocal of \(\cos \theta\) and remains positive in the fourth quadrant. So (\sec\(2\pi-\theta\)=\sec \theta).
Step 2
Why this answer is correct
The correct answer is D. \(\sec \theta\). \(\sec \theta\) is the reciprocal of \(\cos \theta\) and remains positive in the fourth quadrant. So (\sec\(2\pi-\theta\)=\sec \theta).
Step 3
Exam Tip
\(\sec \theta\) \(\cos \theta\) का व्युत्क्रम है और चौथे चतुर्थांश में धनात्मक रहता है। इसलिए (\sec\(2\pi-\theta\)=\sec \theta) है।
\(\cosec \theta\) is the reciprocal of \(\sin \theta\) and is negative in the fourth quadrant. Therefore the value is \(-\cosec \theta\).
Step 2
Why this answer is correct
The correct answer is A. -\(\cosec \theta\). \(\cosec \theta\) is the reciprocal of \(\sin \theta\) and is negative in the fourth quadrant. Therefore the value is \(-\cosec \theta\).
Step 3
Exam Tip
\(\cosec \theta\) \(\sin \theta\) का व्युत्क्रम है और चौथे चतुर्थांश में ऋणात्मक होता है। इसलिए मान \(-\cosec \theta\) है।
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.
Step 2
Why this answer is correct
The correct answer is B. (1). 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.
Step 3
Exam Tip
\(\sin \theta\) का मान ([-1,1]) में होता है इसलिए \(\sin^2 \theta\) का अधिकतम मान (1) है। वर्ग करने पर मान ऋणात्मक नहीं रहता।
Putting \(\sin \theta=0\) in \(\sin^2 \theta+\cos^2 \theta=1\) gives \(\cos^2 \theta=1\). Use the identity in such questions.
Step 2
Why this answer is correct
The correct answer is A. (1). Putting \(\sin \theta=0\) in \(\sin^2 \theta+\cos^2 \theta=1\) gives \(\cos^2 \theta=1\). Use the identity in such questions.
Step 3
Exam Tip
\(\sin^2 \theta+\cos^2 \theta=1\) में \(\sin \theta=0\) रखने पर \(\cos^2 \theta=1\) मिलता है। ऐसे प्रश्नों में पहचान सूत्र लगाएँ।
Putting \(\cos \theta=0\) in \(\sin^2 \theta+\cos^2 \theta=1\) gives \(\sin^2 \theta=1\). Substitute the known value in the formula.
Step 2
Why this answer is correct
The correct answer is B. (1). Putting \(\cos \theta=0\) in \(\sin^2 \theta+\cos^2 \theta=1\) gives \(\sin^2 \theta=1\). Substitute the known value in the formula.
Step 3
Exam Tip
\(\sin^2 \theta+\cos^2 \theta=1\) में \(\cos \theta=0\) रखने पर \(\sin^2 \theta=1\) आता है। ज्ञात मान को सूत्र में रखें।
\(\1+\tan^2 \theta=\sec^2 \theta\) and \(\frac{1}{\sec^2 \theta}=\cos^2 \theta\). Be careful while taking reciprocals.
Step 2
Why this answer is correct
The correct answer is C. \(\cos^2 \theta\). \(\1+\tan^2 \theta=\sec^2 \theta\) and \(\frac{1}{\sec^2 \theta}=\cos^2 \theta\). Be careful while taking reciprocals.
Step 3
Exam Tip
\(\1+\tan^2 \theta=\sec^2 \theta\) और \(\frac{1}{\sec^2 \theta}=\cos^2 \theta\) होता है। व्युत्क्रम लेते समय सावधानी रखें।
\(\1+\cot^2 \theta=\cosec^2 \theta\) and \(\frac{1}{\cosec^2 \theta}=\sin^2 \theta\). Simplify fractions using identities.
Step 2
Why this answer is correct
The correct answer is D. \(\sin^2 \theta\). \(\1+\cot^2 \theta=\cosec^2 \theta\) and \(\frac{1}{\cosec^2 \theta}=\sin^2 \theta\). Simplify fractions using identities.
Step 3
Exam Tip
\(\1+\cot^2 \theta=\cosec^2 \theta\) और \(\frac{1}{\cosec^2 \theta}=\sin^2 \theta\) होता है। पहचान सूत्र से भिन्न सरल करें।
From \(\sec^2 \theta=1+\tan^2 \theta\), we get \(\sec^2 \theta-1=\tan^2 \theta\). Rearrange the formula into a simple form.
Step 2
Why this answer is correct
The correct answer is A. \(\tan^2 \theta\). From \(\sec^2 \theta=1+\tan^2 \theta\), we get \(\sec^2 \theta-1=\tan^2 \theta\). Rearrange the formula into a simple form.
Step 3
Exam Tip
\(\sec^2 \theta=1+\tan^2 \theta\) से \(\sec^2 \theta-1=\tan^2 \theta\) मिलता है। सूत्र को सरल रूप में बदलें।
From \(\cosec^2 \theta=1+\cot^2 \theta\), \(\cosec^2 \theta-1=\cot^2 \theta\). Practice similar identities in pairs.
Step 2
Why this answer is correct
The correct answer is B. \(\cot^2 \theta\). From \(\cosec^2 \theta=1+\cot^2 \theta\), \(\cosec^2 \theta-1=\cot^2 \theta\). Practice similar identities in pairs.
Step 3
Exam Tip
\(\cosec^2 \theta=1+\cot^2 \theta\) से \(\cosec^2 \theta-1=\cot^2 \theta\) होता है। समान पहचान सूत्रों की जोड़ी बनाकर अभ्यास करें।
Since \(\cosec \theta=\frac{1}{\sin \theta}\), \(\frac{\sin \theta}{\cosec \theta}=\sin^2 \theta\). Convert reciprocal relations into fractions.
Step 2
Why this answer is correct
The correct answer is C. \(\sin^2 \theta\). Since \(\cosec \theta=\frac{1}{\sin \theta}\), \(\frac{\sin \theta}{\cosec \theta}=\sin^2 \theta\). Convert reciprocal relations into fractions.
Step 3
Exam Tip
\(\cosec \theta=\frac{1}{\sin \theta}\) इसलिए \(\frac{\sin \theta}{\cosec \theta}=\sin^2 \theta\) है। व्युत्क्रम संबंध को भिन्न में बदलें।
Since \(\sec \theta=\frac{1}{\cos \theta}\), \(\frac{\cos \theta}{\sec \theta}=\cos^2 \theta\). Writing the reciprocal makes simplification easy.
Step 2
Why this answer is correct
The correct answer is D. \(\cos^2 \theta\). Since \(\sec \theta=\frac{1}{\cos \theta}\), \(\frac{\cos \theta}{\sec \theta}=\cos^2 \theta\). Writing the reciprocal makes simplification easy.
Step 3
Exam Tip
\(\sec \theta=\frac{1}{\cos \theta}\) इसलिए \(\frac{\cos \theta}{\sec \theta}=\cos^2 \theta\) होता है। व्युत्क्रम लिखने से सरलीकरण आसान होता है।
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.
Step 2
Why this answer is correct
The correct answer is D. \(\sin \theta\). 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.
Step 3
Exam Tip
\(\tan \theta=\frac{\sin \theta}{\cos \theta}\) और \(\sec \theta=\frac{1}{\cos \theta}\) रखने पर उत्तर \(\sin \theta\) मिलता है। भाग को भिन्न में बदलकर काटें।
Using \(\cot \theta=\frac{\cos \theta}{\sin \theta}\) and \(\cosec \theta=\frac{1}{\sin \theta}\), we get \(\cos \theta\). Writing standard forms is the safest method.
Step 2
Why this answer is correct
The correct answer is A. \(\cos \theta\). 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.
Step 3
Exam Tip
\(\cot \theta=\frac{\cos \theta}{\sin \theta}\) और \(\cosec \theta=\frac{1}{\sin \theta}\) रखने पर \(\cos \theta\) मिलता है। मानक रूप लिखना सबसे सुरक्षित तरीका है।
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.
Step 2
Why this answer is correct
The correct answer is B. \(\cosec \theta\). 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.
Step 3
Exam Tip
\(\sec \theta=\frac{1}{\cos \theta}\) और \(\tan \theta=\frac{\sin \theta}{\cos \theta}\) से \(\frac{1}{\sin \theta}=\cosec \theta\) मिलता है। भिन्नों को सावधानी से सरल करें।
From \(\cosec \theta=\frac{1}{\sin \theta}\) and \(\cot \theta=\frac{\cos \theta}{\sin \theta}\), we get \(\frac{1}{\cos \theta}=\sec \theta\). Write the final answer as a standard function.
Step 2
Why this answer is correct
The correct answer is A. \(\sec \theta\). 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.
Step 3
Exam Tip
\(\cosec \theta=\frac{1}{\sin \theta}\) और \(\cot \theta=\frac{\cos \theta}{\sin \theta}\) से \(\frac{1}{\cos \theta}=\sec \theta\) मिलता है। अंतिम उत्तर को मानक फलन में लिखें।
