KC Sinha Mathematics Solution Class 11 Chapter 5 Trigonometric Functions and Their graphs Exercise 5.2

 Exercise 5.2

Page no 5.18

Type 1

Question 1 

Which trigonometric ratios are negative for the following angles :

(i) $315^{\circ}$

(ii) $-210^{\circ}$

(iii) $\left(-\frac{5 \pi}{6}\right)^{c}$

Page no 5.19

Question 2

Find the values of the other five trigonometric functions in the following

(i) $\cos \theta=-\frac{8}{17}, \pi<\theta<\frac{3 \pi}{2}$

(ii) $\cos \theta=-\frac{1}{2}, \theta$ lies is the third quadrant

(iii) $\cot \theta=-\frac{12}{5}, \frac{\pi}{2}<\theta<\pi$

(iv) $\sec \theta=\frac{13}{5}, \frac{3 \pi}{2}<\theta<2 \pi$

(v) $\tan \theta=\frac{4}{3}, \pi<\theta<\frac{3 \pi}{2}$

(vi) $\cot x=-\frac{5}{12}, \frac{\pi}{2}<x<\pi$

Question 3

If $\cos \theta=\frac{3}{5}$ and $\theta$ lies in the fourth quadrant find the values of $\operatorname{cosec} \theta+\cot \theta$

Question 4

If $\tan \theta=-\frac{4}{3}, \frac{3 \pi}{2}<\theta<2 \pi$, find the value of $9 \sec ^{2} \theta-4 \cot \theta$.

Question 5

If $\sec \theta=\sqrt{2}$ and $\frac{3 \pi}{2}<\theta<2 \pi$, find the value of $\frac{1+\tan \theta+\operatorname{cosec} \theta}{1+\cot \theta-\operatorname{cosec} \theta}$.

Question 6

If $\cos \theta=-\frac{3}{5}$ and $\pi<\theta<\frac{3 \pi}{2}$, find the value of $\frac{\sec \theta-\tan \theta}{\operatorname{cosec} \theta+\cot \theta}$.

Question 7

If $\sin \alpha=\frac{3}{5}, \tan \beta=\frac{1}{2}$ and $\frac{\pi}{2}<\alpha<\pi<\beta<\frac{3 \pi}{2}$, show that $8 \tan \alpha-\sqrt{5} \sec \beta=-\frac{7}{2}$

Type 2

Question 8

Show that $\sin ^{2} \theta+\operatorname{cosec}^{2} \theta \geq 2$.

Question 9

Show that the equation $\sin \theta=x+\frac{1}{x}$ is impossible if $x$ is real.

Question 10

Prove that $\sin ^{2} \theta=\frac{(x+y)^{2}}{4 x y}$ is possible for real values of $x$ and $y$ only when $x=y$ and $x \neq 0$.

[HOTS|

Question 11

Show that $\sin ^{2} \theta=\frac{x^{2}+y^{2}}{2 x y}$ is possible for real values of $x$ and $y$ only when $x=y \neq 0$

[HOTS]

Question 12

Prove that $\sec \theta+\cos \theta \neq \frac{3}{2}$

Question 13

Can $6 \sin ^{2} \theta-7 \sin \theta+2=0$ for any real value of $\theta$ ?

Type 3

Question 14

Find the values of the following trigonometric functions :

(i) $\sin 1830^{\circ}$

(ii) $\sin 765^{\circ}$

(iii) $\cos \left(-1710^{\circ}\right)$

(iv) $\sin \left(-\frac{11 \pi}{3}\right)$

(v) $\sin \frac{31 \pi}{3}$

(vi) $\cot \left(-\frac{15 \pi}{4}\right)$

(vii) $\cot \frac{13 \pi}{3}$

Question 15

Find the sign of the following :

(i) $\tan 2500^{\circ}$

(ii) $\cos \left(-2000^{\circ}\right)$

(iii) $\operatorname{cosec}\left(-3020^{\circ}\right)$