Because for complementary angles ( \sin\left\(\frac{\pi}{2}-\theta\right\)=\cos \theta) and ( \cos\left\(\frac{\pi}{2}-\theta\right\)=\sin \theta). In exams remember co-function formulas.
Step 2
Why this answer is correct
The correct answer is B. \(\sin \theta+\cos \theta\). Because for complementary angles ( \sin\left\(\frac{\pi}{2}-\theta\right\)=\cos \theta) and ( \cos\left\(\frac{\pi}{2}-\theta\right\)=\sin \theta). In exams remember co-function formulas.
Step 3
Exam Tip
क्योंकि पूरक कोणों में (\sin\left\(\frac{\pi}{2}-\theta\right\)=\cos \theta) और (\cos\left\(\frac{\pi}{2}-\theta\right\)=\sin \theta) होता है। परीक्षा में co-function सूत्र याद रखें।
A. \(\sec \theta\) धनात्मक और \(\cosec \theta\) ऋणात्मक/\(\sec \theta\) positive and \(\cosec \theta\) negative
Step 1
Concept
In the fourth quadrant, \(\cos \theta\) is positive and \(\sin \theta\) is negative, so their reciprocals have the same signs. In exams decide the signs of basic functions first.
Step 2
Why this answer is correct
The correct answer is A. \(\sec \theta\) धनात्मक और \(\cosec \theta\) ऋणात्मक / \(\sec \theta\) positive and \(\cosec \theta\) negative. In the fourth quadrant, \(\cos \theta\) is positive and \(\sin \theta\) is negative, so their reciprocals have the same signs. In exams decide the signs of basic functions first.
Step 3
Exam Tip
चतुर्थ चतुर्थांश में \(\cos \theta\) धनात्मक और \(\sin \theta\) ऋणात्मक होता है, इसलिए उनके व्युत्क्रमों के चिन्ह भी वैसे ही होंगे। परीक्षा में पहले मूल फलनों के चिन्ह तय करें।
The fundamental period of \(\tan \theta\) is \(\pi\), and \(5\pi\) is a multiple of the period, so the value does not change. In exams simplify tangent by removing multiples of \(\pi\).
Step 2
Why this answer is correct
The correct answer is C. \(\tan \theta\). The fundamental period of \(\tan \theta\) is \(\pi\), and \(5\pi\) is a multiple of the period, so the value does not change. In exams simplify tangent by removing multiples of \(\pi\).
Step 3
Exam Tip
\(\tan \theta\) का मूल period \(\pi\) है और \(5\pi\) period का गुणज है, इसलिए मान नहीं बदलता। परीक्षा में tangent में \(\pi\) के गुणज को हटाकर सरल करें।
\( \cos^2 \theta=1-\sin^2 \theta=\frac{144}{169}\), and in the first quadrant it is positive. In exams do not forget quadrant sign.
Step 2
Why this answer is correct
The correct answer is A. \(\frac{12}{13}\). \( \cos^2 \theta=1-\sin^2 \theta=\frac{144}{169}\), and in the first quadrant it is positive. In exams do not forget quadrant sign.
Step 3
Exam Tip
\(\cos^2 \theta=1-\sin^2 \theta=\frac{144}{169}\) और प्रथम चतुर्थांश में मान धनात्मक है। परीक्षा में quadrant sign न भूलें।
\( \sin^2 \theta=1-\cos^2 \theta=\frac{16}{25}\), and \(\sin \theta\) is positive in the second quadrant. In exams ASTC rule helps.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{4}{5}\). \( \sin^2 \theta=1-\cos^2 \theta=\frac{16}{25}\), and \(\sin \theta\) is positive in the second quadrant. In exams ASTC rule helps.
Step 3
Exam Tip
\(\sin^2 \theta=1-\cos^2 \theta=\frac{16}{25}\) और द्वितीय चतुर्थांश में \(\sin \theta\) धनात्मक है। परीक्षा में ASTC नियम काम आता है।
\( \pi-\theta\) lies in the second quadrant and sine remains positive. In exams remember allied angle formulas.
Step 2
Why this answer is correct
The correct answer is C. \(\sin \theta\). \( \pi-\theta\) lies in the second quadrant and sine remains positive. In exams remember allied angle formulas.
Step 3
Exam Tip
\(\pi-\theta\) द्वितीय चतुर्थांश में आता है और sine धनात्मक रहता है। परीक्षा में allied angle formula याद रखें।
\( \pi-\theta\) is in the second quadrant and tangent is negative. In exams pay special attention to the sign of tangent.
Step 2
Why this answer is correct
The correct answer is B. \(-\tan \theta\). \( \pi-\theta\) is in the second quadrant and tangent is negative. In exams pay special attention to the sign of tangent.
Step 3
Exam Tip
\(\pi-\theta\) द्वितीय चतुर्थांश में है और tangent ऋणात्मक होता है। परीक्षा में tangent के sign पर विशेष ध्यान दें।
\( \pi+\theta\) is in the third quadrant where sine is negative. In exams identifying allied angles is important.
Step 2
Why this answer is correct
The correct answer is B. \(-\sin \theta\). \( \pi+\theta\) is in the third quadrant where sine is negative. In exams identifying allied angles is important.
Step 3
Exam Tip
\(\pi+\theta\) तृतीय चतुर्थांश में है जहां sine ऋणात्मक होता है। परीक्षा में allied angle पहचानना जरूरी है।
The period of \( \tan \theta\) is \( \pi\), so ( \tan\(\pi+\theta\)=\tan \theta). In exams remember the periodicity of tangent.
Step 2
Why this answer is correct
The correct answer is B. \(\tan \theta\). The period of \( \tan \theta\) is \( \pi\), so ( \tan\(\pi+\theta\)=\tan \theta). In exams remember the periodicity of tangent.
Step 3
Exam Tip
\(\tan \theta\) का period \(\pi\) है इसलिए (\tan\(\pi+\theta\)=\tan \theta)। परीक्षा में tangent की periodicity याद रखें।
\(2\pi-\theta\) is in the fourth quadrant where tangent is negative. In exams derive tangent from sine and cosine signs.
Step 2
Why this answer is correct
The correct answer is B. \(-\tan \theta\). \(2\pi-\theta\) is in the fourth quadrant where tangent is negative. In exams derive tangent from sine and cosine signs.
Step 3
Exam Tip
\(2\pi-\theta\) चतुर्थ चतुर्थांश में है जहां tangent ऋणात्मक होता है। परीक्षा में sine और cosine के sign से tangent निकालें।
\( \sin \theta\) is zero at points like (0), \(\pi\), and \(2\pi\). In exams write the general solution with \(n\in\mathbb{Z}\).
Step 2
Why this answer is correct
The correct answer is A. \(\theta=n\pi\). \( \sin \theta\) is zero at points like (0), \(\pi\), and \(2\pi\). In exams write the general solution with \(n\in\mathbb{Z}\).
Step 3
Exam Tip
\(\sin \theta\) शून्य (0), \(\pi\), \(2\pi\) जैसे बिंदुओं पर होता है। परीक्षा में \(n\in\mathbb{Z}\) मानकर सामान्य हल लिखें।
\( \cos \theta\) is zero at odd multiples of \( \frac{\pi}{2}\). In exams remember the pattern ( \frac{(2n+1)\pi}{2}).
Step 2
Why this answer is correct
The correct answer is B. (\theta=\frac{(2n+1)\pi}{2}). \( \cos \theta\) is zero at odd multiples of \( \frac{\pi}{2}\). In exams remember the pattern ( \frac{(2n+1)\pi}{2}).
Step 3
Exam Tip
\(\cos \theta\) शून्य odd multiples of \(\frac{\pi}{2}\) पर होता है। परीक्षा में (\frac{(2n+1)\pi}{2}) पैटर्न याद रखें।
\( \tan \theta=0\) occurs when \( \sin \theta=0\) and \( \cos \theta\neq0\). In exams use period \( \pi\) for tangent.
Step 2
Why this answer is correct
The correct answer is A. \(\theta=n\pi\). \( \tan \theta=0\) occurs when \( \sin \theta=0\) and \( \cos \theta\neq0\). In exams use period \( \pi\) for tangent.
Step 3
Exam Tip
\(\tan \theta=0\) तब होता है जब \(\sin \theta=0\) और \(\cos \theta\neq0\)। परीक्षा में tangent का period \(\pi\) रखें।
The combined solution of \( \sin \theta=\sin \alpha\) is ( \theta=n\pi+(-1)^n\alpha). In exams learn to write two cases in one formula.
Step 2
Why this answer is correct
The correct answer is A. (\theta=n\pi+(-1)^n\alpha). The combined solution of \( \sin \theta=\sin \alpha\) is ( \theta=n\pi+(-1)^n\alpha). In exams learn to write two cases in one formula.
Step 3
Exam Tip
\(\sin \theta=\sin \alpha\) का संयुक्त हल (\theta=n\pi+(-1)^n\alpha) होता है। परीक्षा में दो अलग हलों को एक सूत्र में लिखना सीखें।
For \( \cos \theta=\cos \alpha\), angles occur in both directions. In exams do not forget \( \pm\alpha\).
Step 2
Why this answer is correct
The correct answer is B. \(\theta=2n\pi\pm\alpha\). For \( \cos \theta=\cos \alpha\), angles occur in both directions. In exams do not forget \( \pm\alpha\).
Step 3
Exam Tip
\(\cos \theta=\cos \alpha\) के लिए दोनों दिशाओं में कोण मिलते हैं। परीक्षा में \(\pm\alpha\) लगाना न भूलें।
The period of \( \tan \theta\) is \( \pi\), so the solution is \( \theta=n\pi+\alpha\). In exams use period \( \pi\) for tangent equations.
Step 2
Why this answer is correct
The correct answer is B. \(\theta=n\pi+\alpha\). The period of \( \tan \theta\) is \( \pi\), so the solution is \( \theta=n\pi+\alpha\). In exams use period \( \pi\) for tangent equations.
Step 3
Exam Tip
\(\tan \theta\) का period \(\pi\) है इसलिए हल \(\theta=n\pi+\alpha\) होगा। परीक्षा में tangent equations में \(\pi\) period लगाएं।
At \( \frac{\pi}{2}-\theta\), tangent changes to its co-function cotangent. In exams remember complementary angle property.
Step 2
Why this answer is correct
The correct answer is C. \(\cot \theta\). At \( \frac{\pi}{2}-\theta\), tangent changes to its co-function cotangent. In exams remember complementary angle property.
Step 3
Exam Tip
\(\frac{\pi}{2}-\theta\) पर tangent का co-function cotangent होता है। परीक्षा में complementary angle property याद रखें।
At \( \frac{\pi}{2}-\theta\), cotangent changes to the co-function tangent. In exams remember reciprocal function pairs.
Step 2
Why this answer is correct
The correct answer is D. \(\tan \theta\). At \( \frac{\pi}{2}-\theta\), cotangent changes to the co-function tangent. In exams remember reciprocal function pairs.
Step 3
Exam Tip
\(\frac{\pi}{2}-\theta\) पर cotangent का co-function tangent होता है। परीक्षा में reciprocal function pair याद रखें।
From \( \sin \theta=\cos \theta\), \( \tan \theta=1\), and in the first quadrant \( \theta=\frac{\pi}{4}\). In exams divide by \( \cos \theta\) only when it is nonzero.
Step 2
Why this answer is correct
The correct answer is C. \(\frac{\pi}{4}\). From \( \sin \theta=\cos \theta\), \( \tan \theta=1\), and in the first quadrant \( \theta=\frac{\pi}{4}\). In exams divide by \( \cos \theta\) only when it is nonzero.
Step 3
Exam Tip
\(\sin \theta=\cos \theta\) से \(\tan \theta=1\) और प्रथम चतुर्थांश में \(\theta=\frac{\pi}{4}\) है। परीक्षा में दोनों तरफ \(\cos \theta\) से भाग तभी दें जब वह शून्य न हो।
The reference angle for \( \sin \theta=\frac{1}{2}\) is \( \frac{\pi}{6}\), and sine is positive in the first and second quadrants. In exams choose only answers in the given interval.
Step 2
Why this answer is correct
The correct answer is A. \(\frac{\pi}{6},\frac{5\pi}{6}\). The reference angle for \( \sin \theta=\frac{1}{2}\) is \( \frac{\pi}{6}\), and sine is positive in the first and second quadrants. In exams choose only answers in the given interval.
Step 3
Exam Tip
\(\sin \theta=\frac{1}{2}\) के reference angle \(\frac{\pi}{6}\) हैं और sine प्रथम व द्वितीय चतुर्थांश में धनात्मक है। परीक्षा में दिए interval में ही उत्तर चुनें।
For \( \cos \theta=\frac{1}{2}\), the reference angle is \( \frac{\pi}{3}\), and cosine is positive in the first and fourth quadrants. In exams write both solutions up to \(2\pi\).
Step 2
Why this answer is correct
The correct answer is A. \(\frac{\pi}{3},\frac{5\pi}{3}\). For \( \cos \theta=\frac{1}{2}\), the reference angle is \( \frac{\pi}{3}\), and cosine is positive in the first and fourth quadrants. In exams write both solutions up to \(2\pi\).
Step 3
Exam Tip
\(\cos \theta=\frac{1}{2}\) के लिए reference angle \(\frac{\pi}{3}\) है और cosine प्रथम व चतुर्थ चतुर्थांश में धनात्मक है। परीक्षा में \(2\pi\) तक के दोनों हल लिखें।
The reference angle for \( \tan \theta=\sqrt{3}\) is \( \frac{\pi}{3}\), and in \(0<\theta<\pi\) tangent is positive in the first quadrant. In exams choose the quadrant by sign.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{\pi}{3}\). The reference angle for \( \tan \theta=\sqrt{3}\) is \( \frac{\pi}{3}\), and in \(0<\theta<\pi\) tangent is positive in the first quadrant. In exams choose the quadrant by sign.
Step 3
Exam Tip
\(\tan \theta=\sqrt{3}\) का reference angle \(\frac{\pi}{3}\) है और \(0<\theta<\pi\) में tangent प्रथम चतुर्थांश में धनात्मक है। परीक्षा में sign के अनुसार quadrant चुनें।
In \( \tan \theta=\frac{4}{3}\), opposite is (4), adjacent is (3), and hypotenuse is (5). In exams identify Pythagorean triplets.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{4}{5}\). In \( \tan \theta=\frac{4}{3}\), opposite is (4), adjacent is (3), and hypotenuse is (5). In exams identify Pythagorean triplets.
Step 3
Exam Tip
\(\tan \theta=\frac{4}{3}\) में opposite (4), adjacent (3), hypotenuse (5) होगा। परीक्षा में Pythagorean triplet पहचानें।
\(In ( \cot \theta=\frac{12}{5}), adjacent is (12), opposite is (5), and hypotenuse is (13). In exams use ( \cos \theta=\frac{\)adjacent}{hypotenuse}).
Step 2
Why this answer is correct
\(The correct answer is B. (\frac{12}{13}). In ( \cot \theta=\frac{12}{5}), adjacent is (12), opposite is (5), and hypotenuse is (13). In exams use ( \cos \theta=\frac{\)adjacent}{hypotenuse}).
Step 3
Exam Tip
\(\cot \theta=\frac{12}{5}\) में adjacent (12), opposite (5), hypotenuse (13) है। \(परीक्षा में (\cos \theta=\frac{\)adjacent}{hypotenuse}) लगाएं।
\( \sec \theta=\frac{1}{\cos \theta}\), so \( \cos \theta=\frac{1}{2}\). In exams remember reciprocal relations.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{1}{2}\). \( \sec \theta=\frac{1}{\cos \theta}\), so \( \cos \theta=\frac{1}{2}\). In exams remember reciprocal relations.
Step 3
Exam Tip
\(\sec \theta=\frac{1}{\cos \theta}\), इसलिए \(\cos \theta=\frac{1}{2}\) है। परीक्षा में reciprocal relations याद रखें।
\( \cosec \theta=\frac{1}{\sin \theta}\), so \( \sin \theta=\frac{2}{5}\). In exams remember cosecant and sine are reciprocals.
Step 2
Why this answer is correct
The correct answer is C. \(\frac{2}{5}\). \( \cosec \theta=\frac{1}{\sin \theta}\), so \( \sin \theta=\frac{2}{5}\). In exams remember cosecant and sine are reciprocals.
Step 3
Exam Tip
\(\cosec \theta=\frac{1}{\sin \theta}\), इसलिए \(\sin \theta=\frac{2}{5}\) है। परीक्षा में cosecant और sine reciprocal हैं।
Because \( \cos 2\theta=\cos^2 \theta-\sin^2 \theta\), the given form is \(-\cos 2\theta\). In exams read double angle identities in reverse too.
Step 2
Why this answer is correct
The correct answer is B. \(-\cos 2\theta\). Because \( \cos 2\theta=\cos^2 \theta-\sin^2 \theta\), the given form is \(-\cos 2\theta\). In exams read double angle identities in reverse too.
Step 3
Exam Tip
क्योंकि \(\cos 2\theta=\cos^2 \theta-\sin^2 \theta\), इसलिए दिया गया रूप \(-\cos 2\theta\) है। परीक्षा में double angle identities को उल्टा भी पढ़ें।
In the first quadrant, the maximum of \( \sin \theta+\cos \theta\) is \( \sqrt{2}\) at \( \theta=\frac{\pi}{4}\). In exams use symmetry.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{\pi}{4}\). In the first quadrant, the maximum of \( \sin \theta+\cos \theta\) is \( \sqrt{2}\) at \( \theta=\frac{\pi}{4}\). In exams use symmetry.
Step 3
Exam Tip
प्रथम चतुर्थांश में \(\sin \theta+\cos \theta\) का अधिकतम \(\sqrt{2}\) \(\theta=\frac{\pi}{4}\) पर होता है। परीक्षा में symmetry का उपयोग करें।
( \sin \theta+\cos \theta=\sqrt{2}\sin\left\(\theta+\frac{\pi}{4}\right\)), so the maximum is \( \sqrt{2}\). In exams use maximum of \(a\sin x+b\cos x\) as \( \sqrt{a^2+b^2}\).
Step 2
Why this answer is correct
The correct answer is B. \(\sqrt{2}\). ( \sin \theta+\cos \theta=\sqrt{2}\sin\left\(\theta+\frac{\pi}{4}\right\)), so the maximum is \( \sqrt{2}\). In exams use maximum of \(a\sin x+b\cos x\) as \( \sqrt{a^2+b^2}\).
Step 3
Exam Tip
(\sin \theta+\cos \theta=\sqrt{2}\sin\left\(\theta+\frac{\pi}{4}\right\)), इसलिए अधिकतम \(\sqrt{2}\) है। परीक्षा में \(a\sin x+b\cos x\) का maximum \(\sqrt{a^2+b^2}\) लें।
The amplitude of \( \sin \theta-\cos \theta\) is ( \sqrt{12+(-1)2}=\sqrt{2}). In exams the minimum is the negative amplitude.
Step 2
Why this answer is correct
The correct answer is A. \(-\sqrt{2}\). The amplitude of \( \sin \theta-\cos \theta\) is ( \sqrt{12+(-1)2}=\sqrt{2}). In exams the minimum is the negative amplitude.
Step 3
Exam Tip
\(\sin \theta-\cos \theta\) का amplitude (\sqrt{12+(-1)2}=\sqrt{2}) है। परीक्षा में minimum हमेशा negative amplitude होगा।
The amplitude is ( \sqrt{52+(-12)2}=13), so the minimum is (-13). In exams maximum is positive amplitude and minimum is negative amplitude.
Step 2
Why this answer is correct
The correct answer is B. (-13). The amplitude is ( \sqrt{52+(-12)2}=13), so the minimum is (-13). In exams maximum is positive amplitude and minimum is negative amplitude.
Step 3
Exam Tip
Amplitude (\sqrt{52+(-12)2}=13) है, इसलिए न्यूनतम (-13) होगा। परीक्षा में maximum positive और minimum negative amplitude होता है।
B. \(\sin \theta\) ऋणात्मक और \(\cos \theta\) ऋणात्मक/\(\sin \theta\) negative and \(\cos \theta\) negative
Step 1
Concept
In the third quadrant, both sine and cosine are negative. In exams use the ASTC rule to decide signs quickly.
Step 2
Why this answer is correct
The correct answer is B. \(\sin \theta\) ऋणात्मक और \(\cos \theta\) ऋणात्मक / \(\sin \theta\) negative and \(\cos \theta\) negative. In the third quadrant, both sine and cosine are negative. In exams use the ASTC rule to decide signs quickly.
Step 3
Exam Tip
तृतीय चतुर्थांश में sine और cosine दोनों ऋणात्मक होते हैं। परीक्षा में ASTC नियम से signs जल्दी तय करें।
The period of \( \sin k\theta\) is \( \frac{2\pi}{k}\), so here it is \( \frac{2\pi}{3}\). In exams use the coefficient in the period formula.
Step 2
Why this answer is correct
The correct answer is A. \(\frac{2\pi}{3}\). The period of \( \sin k\theta\) is \( \frac{2\pi}{k}\), so here it is \( \frac{2\pi}{3}\). In exams use the coefficient in the period formula.
Step 3
Exam Tip
\(\sin k\theta\) का period \(\frac{2\pi}{k}\) होता है, इसलिए यहां \(\frac{2\pi}{3}\) मिलेगा। परीक्षा में coefficient को period formula में लगाएं।
\( \cos \theta=-1\) occurs at odd multiples of \( \pi\). In exams distinguish the solutions of \( \cos \theta=1\) and \( \cos \theta=-1\).
Step 2
Why this answer is correct
The correct answer is B. (\theta=(2n+1)\pi). \( \cos \theta=-1\) occurs at odd multiples of \( \pi\). In exams distinguish the solutions of \( \cos \theta=1\) and \( \cos \theta=-1\).
Step 3
Exam Tip
\(\cos \theta=-1\) विषम गुणजों of \(\pi\) पर होता है। परीक्षा में \(\cos \theta=1\) और \(\cos \theta=-1\) के हल अलग पहचानें।
The fundamental period of \( \cos k\theta\) is \( \frac{2\pi}{k}\), so here it is \( \frac{\pi}{2}\). In exams put the coefficient in the denominator.
Step 2
Why this answer is correct
The correct answer is A. \(\frac{\pi}{2}\). The fundamental period of \( \cos k\theta\) is \( \frac{2\pi}{k}\), so here it is \( \frac{\pi}{2}\). In exams put the coefficient in the denominator.
Step 3
Exam Tip
\(\cos k\theta\) का मूल period \(\frac{2\pi}{k}\) होता है, इसलिए यहां \(\frac{\pi}{2}\) मिलेगा। परीक्षा में coefficient को denominator में रखें।
