Chapter 12

Which of the following statements are correct/incorrect? i. \(\sin \theta=-\frac{1}{5} .\\{\) Ans. correct \(\\}\) ii. \(\cos \theta=1 .\\{\) Ans. correct \(\\}\) iii. \(\sec \theta=\frac{1}{2}\). \\{Ans. incorrect\\} iv. \(\tan \theta=20\). \\{Ans. correct\\}

Find the maximum and minimum values of \(7 \cos \theta+24 \sin \theta\).

Show that the maximum and minimum values of \(8 \cos \theta-15 \sin \theta\) are 17 and \(-17\) respectively.

Find the maximum and minimum values of \(3 \cos x+4 \sin x+5\).

For what value of \(x\) in the interval \(\left(0, \frac{\pi}{2}\right)\), the maximum value of \(\sin \left(x+\frac{\pi}{6}\right)+\cos \left(x+\frac{\pi}{6}\right)\) is attained?

Show that \(|\sin x+\cos x| \leq \sqrt{2}\).

Prove that \(5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3\) lies between \(-4\) and 10 .

Find \(a\) and \(b\) such that the inequality \(a \leq 3 \cos x+5 \sin \left(x-\frac{\pi}{6}\right) \leq b\) holds good for all \(x\). \\{Ans.

Find the maximum and minimum values of \(6 \sin x \cos x+4 \cos 2 x\).

Show that for all values of \(\theta\), the expression \(a \sin ^{2} \theta+b \sin \theta \cos \theta+c \cos ^{2} \theta\) lies between \(\frac{a+c}{2}-\frac{\sqrt{b^{2}+(a-c)^{2}}}{2}\) and \(\frac{a+c}{2}+\frac{\sqrt{b^{2}+(a-c)^{2}}}{2}\)

Express \(6 \cos ^{2} \alpha+8 \sin \alpha \cos \alpha\) as \(A+B \cos (2 \alpha-\beta)\) and hence show that the greatest and the least values of the expression are 8 and \(-2\) respectively.

Find the maximum and minimum values of \(\cos 2 x+9 \sin x\).

Prove that \(-4 \leq \cos 2 x+3 \sin x \leq \frac{17}{8}\)

If \(a \leq \cos 2 x+5 \sin x+6 \leq b\), find \(a\) and \(b\).

Show that the value of \(\sec ^{2} \theta+\cos ^{2} \theta\) is never less than 2 .

Prove that \(\frac{\sin \alpha}{\sin \beta}+\frac{\sin \beta}{\sin \alpha}=\frac{(\sin \alpha-\sin \beta)^{2}}{\sin \alpha \sin \beta}+2\). Hence deduce that if \(0<\alpha, \beta<\pi, \frac{\sin \alpha}{\sin \beta}+\frac{\sin \beta}{\sin \alpha} \geq 2\).

Find the greatest and least values of \(\cos A \cos B\) when \(A+B=90^{\circ}\).

If \(A>0, B>0\) and \(A+B=\frac{\pi}{3}\), find the maximum value of \(\tan A \tan B\).

Prove that \(\frac{\cot 3 x}{\cot x}\) never lies between \(\frac{1}{3}\) and 3 .

Prove that \(\tan \left(x+\frac{\pi}{6}\right) \cot x\) cannot lie between \(\frac{1}{3}\) and 3 .

Show that \(\frac{3+\cos \theta}{\sin \theta}\) cannot have any value between \(-2 \sqrt{2}\) and \(2 \sqrt{2}\). What are the limits of \(\frac{\sin \theta}{3+\cos \theta}\).

Prove that the expression \(\cos \theta\left(\sin \theta+\sqrt{\sin ^{2} \alpha+\sin ^{2} \theta}\right)\) lies between \(-\sqrt{1+\sin ^{2} \alpha}\) and \(\sqrt{1+\sin ^{2} \alpha}\), \(0<\theta<\frac{\pi}{2}\)

For all \(\theta\) in \(\left[0, \frac{\pi}{2}\right]\) show that \(\cos (\sin \theta) \geq \sin (\cos \theta)\).

If \(0<\theta<\pi\), prove that \(\cot \frac{\theta}{4}-\cot \theta>2\) and \(\cot \frac{\theta}{2}-\cot \theta \geq 1\).

$$ \left\\{\begin{array}{l} \cos \theta=-\frac{1}{\sqrt{2}} \\ \tan \theta=1 \end{array}\right\\} $$

\(\tan 3 x=1\)

\(\frac{\cos x}{1+\cos 2 x}=0\)

\(\frac{\sin x+\cos x}{\cos 2 x}=0\)

\(\cos x \tan 3 x=0\)

\(\sin 4 x \cos x \tan 2 x=0\)

\((1+\cos x)\left(\frac{1}{\sin x}-1\right)=0\)

\((1+\cos x) \tan \frac{x}{2}=0\)

\(2 \sin ^{2} \theta-3 \sin \theta-2=0\)

\(\sin ^{2} 3 x-5 \sin 3 x+4=0\)

\(8 \sec ^{2} \theta-6 \sec \theta+1=0\)

\(\tan ^{3} x+\tan ^{2} x-3 \tan x=3\)

\(8 \cos ^{4} x-8 \cos ^{2} x-\cos x+1=0\)

\(2 \sin ^{3} x-\cos 2 x-\sin x=0\)

\(2 \cos ^{2} x+5 \sin x-4=0\)

\(3 \sin ^{2} 2 x+7 \cos 2 x=3\)

\(2 \cos ^{2} x+\sin x=2\)

\(\sqrt{2} \sin ^{2} x+\cos x=0\)

\(\sin 2 x+\cos 2 x=\sin x+\cos x\)

\(\sqrt{2} \cos 2 x=\cos x+\sin x\)

\(\sin 3 x=\cos 2 x\)

\(\cos 5 x=\sin 15 x\)

\(\sin (5 \pi-x)=\cos (2 x+7 \pi)\)

\(4 \sin ^{2} x+\sin ^{2} 2 x=3\)

\(4 \cos ^{2} 2 x+8 \cos ^{2} x=7\)

\(\sin \left(x+\frac{\pi}{6}\right)+\cos \left(x+\frac{\pi}{3}\right)=1+\cos 2 x\)