Exercise 6.2 — Class 9 Mathematics Solutions

For each of the following right-angled triangles, find the trigonometric ratios:

(i) $\sin \theta$ (ii) $\cos \theta$ (iii) $\tan \theta$ (iv) $\sec \theta$ (v) $\mathrm{cosec}\theta$ (vi) $\cot \varphi$ (vii) $\tan \varphi$ (viii) $\mathrm{cosec}\varphi$ (ix) $\sec \varphi$ (x) $\cos \varphi$

In each triangle $m\angle B={90}^{\circ }$, $m\angle A=\theta$ and $m\angle C=\varphi$:

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=4$, $m\overline{AB}=3$, $m\overline{AC}=5$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\sin \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=4$, $m\overline{AB}=3$, $m\overline{AC}=5$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\cos \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=4$, $m\overline{AB}=3$, $m\overline{AC}=5$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\tan \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=4$, $m\overline{AB}=3$, $m\overline{AC}=5$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\sec \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=4$, $m\overline{AB}=3$, $m\overline{AC}=5$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\mathrm{cosec}\theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=4$, $m\overline{AB}=3$, $m\overline{AC}=5$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\cot \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=4$, $m\overline{AB}=3$, $m\overline{AC}=5$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\tan \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=4$, $m\overline{AB}=3$, $m\overline{AC}=5$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\mathrm{cosec}\varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=4$, $m\overline{AB}=3$, $m\overline{AC}=5$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\sec \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=4$, $m\overline{AB}=3$, $m\overline{AC}=5$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\cos \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=8$, $m\overline{AB}=15$, $m\overline{AC}=17$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\sin \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=8$, $m\overline{AB}=15$, $m\overline{AC}=17$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\cos \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=8$, $m\overline{AB}=15$, $m\overline{AC}=17$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\tan \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=8$, $m\overline{AB}=15$, $m\overline{AC}=17$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\sec \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=8$, $m\overline{AB}=15$, $m\overline{AC}=17$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\mathrm{cosec}\theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=8$, $m\overline{AB}=15$, $m\overline{AC}=17$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\cot \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=8$, $m\overline{AB}=15$, $m\overline{AC}=17$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\tan \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=8$, $m\overline{AB}=15$, $m\overline{AC}=17$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\mathrm{cosec}\varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=8$, $m\overline{AB}=15$, $m\overline{AC}=17$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\sec \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=8$, $m\overline{AB}=15$, $m\overline{AC}=17$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\cos \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=5$, $m\overline{AB}=12$, $m\overline{AC}=13$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\sin \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=5$, $m\overline{AB}=12$, $m\overline{AC}=13$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\cos \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=5$, $m\overline{AB}=12$, $m\overline{AC}=13$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\tan \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=5$, $m\overline{AB}=12$, $m\overline{AC}=13$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\sec \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=5$, $m\overline{AB}=12$, $m\overline{AC}=13$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\mathrm{cosec}\theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=5$, $m\overline{AB}=12$, $m\overline{AC}=13$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\cot \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=5$, $m\overline{AB}=12$, $m\overline{AC}=13$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\tan \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=5$, $m\overline{AB}=12$, $m\overline{AC}=13$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\mathrm{cosec}\varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=5$, $m\overline{AB}=12$, $m\overline{AC}=13$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\sec \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=5$, $m\overline{AB}=12$, $m\overline{AC}=13$, $m\angle A=\theta$ and $m\angle C=\varphi$. Find $\cos \varphi$.

For the following right-angled triangle $ABC$, find the trigonometric ratios for which $m\angle A=\varphi$ and $m\angle C=\theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, $m\angle A=\varphi$ and $m\angle C=\theta$.

(i) $\sin \theta$ (ii) $\cos \theta$ (iii) $\tan \theta$ (iv) $\sin \varphi$ (v) $\cos \varphi$ (vi) $\tan \varphi$

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, $m\angle A=\varphi$ and $m\angle C=\theta$. Find $\sin \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, $m\angle A=\varphi$ and $m\angle C=\theta$. Find $\cos \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, $m\angle A=\varphi$ and $m\angle C=\theta$. Find $\tan \theta$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, $m\angle A=\varphi$ and $m\angle C=\theta$. Find $\sin \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, $m\angle A=\varphi$ and $m\angle C=\theta$. Find $\cos \varphi$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, $m\angle A=\varphi$ and $m\angle C=\theta$. Find $\tan \varphi$.

Considering the adjoining triangle, verify that:

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{AB}=4$, $m\overline{BC}=3$, $m\overline{AC}=5$ and $m\angle A=\theta$.

(i) $\sin \theta \mathrm{cosec}\theta =1$ (ii) $\cos \theta \sec \theta =1$ (iii) $\tan \theta \cot \theta =1$

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{AB}=4$, $m\overline{BC}=3$, $m\overline{AC}=5$ and $m\angle A=\theta$. Verify that $\sin \theta \mathrm{cosec}\theta =1$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{AB}=4$, $m\overline{BC}=3$, $m\overline{AC}=5$ and $m\angle A=\theta$. Verify that $\cos \theta \sec \theta =1$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$, $m\overline{AB}=4$, $m\overline{BC}=3$, $m\overline{AC}=5$ and $m\angle A=\theta$. Verify that $\tan \theta \cot \theta =1$.

Fill in the blanks.

(i) $\sin {30}^{\circ }=\sin ({90}^{\circ }-{60}^{\circ })=$ ______

(ii) $\cos {30}^{\circ }=\cos ({90}^{\circ }-{60}^{\circ })=$ ______

(iii) $\tan {30}^{\circ }=\tan ({90}^{\circ }-{60}^{\circ })=$ ______

(iv) $\tan {60}^{\circ }=\tan ({90}^{\circ }-{30}^{\circ })=$ ______

(v) $\sin {60}^{\circ }=\sin ({90}^{\circ }-{30}^{\circ })=$ ______

(vi) $\cos {60}^{\circ }=\cos ({90}^{\circ }-{30}^{\circ })=$ ______

(vii) $\sin {45}^{\circ }=\sin ({90}^{\circ }-{45}^{\circ })=$ ______

(viii) $\tan {45}^{\circ }=\tan ({90}^{\circ }-{45}^{\circ })=$ ______

(ix) $\cos {45}^{\circ }=\cos ({90}^{\circ }-{45}^{\circ })=$ ______

Fill in the blank:

$\sin {30}^{\circ }=\sin ({90}^{\circ }-{60}^{\circ })=$ ______

Fill in the blank:

$\cos {30}^{\circ }=\cos ({90}^{\circ }-{60}^{\circ })=$ ______

Fill in the blank:

$\tan {30}^{\circ }=\tan ({90}^{\circ }-{60}^{\circ })=$ ______

Fill in the blank:

$\tan {60}^{\circ }=\tan ({90}^{\circ }-{30}^{\circ })=$ ______

Fill in the blank:

$\sin {60}^{\circ }=\sin ({90}^{\circ }-{30}^{\circ })=$ ______

Fill in the blank:

$\cos {60}^{\circ }=\cos ({90}^{\circ }-{30}^{\circ })=$ ______

Fill in the blank:

$\sin {45}^{\circ }=\sin ({90}^{\circ }-{45}^{\circ })=$ ______

Fill in the blank:

$\tan {45}^{\circ }=\tan ({90}^{\circ }-{45}^{\circ })=$ ______

Fill in the blank:

$\cos {45}^{\circ }=\cos ({90}^{\circ }-{45}^{\circ })=$ ______

If in a right-angled triangle $ABC$, $m\angle B={90}^{\circ }$ and $C$ is an acute angle of ${60}^{\circ }$. Also $\sin m\angle A={\displaystyle \frac{a}{b}}$, then find the following trigonometric ratios.

(Figure: $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$.)

(i) ${\displaystyle \frac{m\overline{BC}}{m\overline{AB}}}$ (ii) $\cos {60}^{\circ }$ (iii) $\tan {60}^{\circ }$ (iv) $\mathrm{cosec}{\displaystyle \frac{\pi }{3}}$ (v) $\cot {60}^{\circ }$ (vi) $\sin {30}^{\circ }$ (vii) $\cos {30}^{\circ }$ (viii) $\tan {\displaystyle \frac{\pi }{6}}$ (ix) $\sec {30}^{\circ }$ (x) $\cot {30}^{\circ }$

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$ and $C$ is an acute angle of ${60}^{\circ }$. The sides are $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, and $\sin m\angle A={\displaystyle \frac{a}{b}}$. Find ${\displaystyle \frac{m\overline{BC}}{m\overline{AB}}}$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$ and $C$ is an acute angle of ${60}^{\circ }$. The sides are $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, and $\sin m\angle A={\displaystyle \frac{a}{b}}$. Find $\cos {60}^{\circ }$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$ and $C$ is an acute angle of ${60}^{\circ }$. The sides are $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, and $\sin m\angle A={\displaystyle \frac{a}{b}}$. Find $\tan {60}^{\circ }$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$ and $C$ is an acute angle of ${60}^{\circ }$. The sides are $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, and $\sin m\angle A={\displaystyle \frac{a}{b}}$. Find $\mathrm{cosec}{\displaystyle \frac{\pi }{3}}$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$ and $C$ is an acute angle of ${60}^{\circ }$. The sides are $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, and $\sin m\angle A={\displaystyle \frac{a}{b}}$. Find $\cot {60}^{\circ }$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$ and $C$ is an acute angle of ${60}^{\circ }$. The sides are $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, and $\sin m\angle A={\displaystyle \frac{a}{b}}$. Find $\sin {30}^{\circ }$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$ and $C$ is an acute angle of ${60}^{\circ }$. The sides are $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, and $\sin m\angle A={\displaystyle \frac{a}{b}}$. Find $\cos {30}^{\circ }$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$ and $C$ is an acute angle of ${60}^{\circ }$. The sides are $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, and $\sin m\angle A={\displaystyle \frac{a}{b}}$. Find $\tan {\displaystyle \frac{\pi }{6}}$.

In right-angled triangle $ABC$, $m\angle B={90}^{\circ }$ and $C$ is an acute angle of ${60}^{\circ }$. The sides are $m\overline{BC}=a$, $m\overline{AB}=c$, $m\overline{AC}=b$, and $\sin m\angle A={\displaystyle \frac{a}{b}}$. Find $\sec {30}^{\circ }$.