Trigonometric Ratios of Common Angles

Q.

${\tan }^{-1}\left(\sqrt{3}\right)-{\cot }^{-1}(-\sqrt{3})$ is equal to

1. $0$

2. $2\sqrt{3}$

3. $–\frac{\mathrm{\pi }}{2}$

4. $\mathrm{\pi }$

Q. In triangle ABC, right angled at B, if one angle is ${45}^{\circ }$, find the value of sin A, cosC, cot A and tan C respectively.

1. $2,2,\sqrt{2},\sqrt{2}$
2. $1/\sqrt{2},1/\sqrt{2},1,1$
3. $1,1,1/\sqrt{3},1/\sqrt{3}$
4. $1/\sqrt{3},1/\sqrt{3},1,1$

Q.

What are the $6$ trigonometric ratios ?

Q. If $P=\frac{\sin {300}^{\circ }\cdot \tan {330}^{\circ }\cdot \sec {420}^{\circ }}{\tan {135}^{\circ }\cdot \sin {210}^{\circ }\cdot \sec {315}^{\circ }}$ and $Q=\frac{\sec {480}^{\circ }\cdot \text{cosec}{570}^{\circ }\cdot \tan {330}^{\circ }}{\sin {600}^{\circ }\cdot \cos {660}^{\circ }\cdot \cot {405}^{\circ }},$ then the value of $P$ and $Q$ are respectively

1. $\sqrt{2},-\frac{16}{3}$
2. $\sqrt{2},\frac{16}{3}$
3. $-2,\frac{3}{16}$
4. $\sqrt{2},\frac{3}{16}$

Q.

In triangle ABC, right angled at B, if one angle is ${45}^o$, find the value of sin A, cos C, cot A and tan C respectively.

1. 1, 1, 1/√3, 1/√3

2. 2, 2, √2, √2

3. 1/√2, 1/√2, 1, 1

4. 1/√3, 1/√3, 1, 1

Q. In triangle ABC, right angled at B, Find the expressions of cos A , sin C, tan A and cot C.

1. AB/AC , BC/AC, BC/AB , AB/BC
2. BC/AC , AB/AC, BC/AB , BC/AB
3. AB/AC , AB/AC, BC/AB , BC/AB
4. BC/AC , BC/AC, AB/BC , BC/AB

Q. The numerical value of $\text{cosec}\theta [\frac{1-\cos \theta }{\sin \theta }+\frac{\sin \theta }{1-\cos \theta }]-2{\cot }^2\theta$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q.

If cos A = $\frac{\sqrt{3}}{2}$, then tan 3A =

1. 0

2. $\frac{1}{2}$

3. 1

4. $\infty$

Q. In triangle ABC, right angled at B, if one angle is ${45}^{\circ }$, the values of sin A, cosC, cot A and tan C respectively are .

1. $2,2,\sqrt{2},\sqrt{2}$
2. $\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}},1,1$
3. $1,1,\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$
4. $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},1,1$

Q.

In an equilateral triangle ABC , match the following ratios to the values

$\begin{array}{cc}\text{Trigonometric Ratios}& \text{Values}\\ \left(i\right)tan{60}^o& \left(a\right)\frac{1}{\sqrt{3}}\\ \left(ii\right)cot{30}^o& \left(b\right)\sqrt{3}\\ \left(iii\right)cosec{30}^o& \left(c\right)2\\ \left(iv\right)sec{30}^o& \left(d\right)\frac{2}{\sqrt{3}}\end{array}$

1. (i) -a , (ii) - a , (iii) - c , (iv) - d

2. (i) -d , (ii) - a , (iii) - b , (iv) - d

3. (i) -a , (ii) - b , (iii) - d , (iv) - c

4. (i) -b , (ii) - b , (iii) - c , (iv) - d

Q. In an equilateral triangle ABC , match the following ratios to the values
$\begin{array}{cc}Trigonometricratios& Values\\ \left(i\right)tan{60}^{\circ }& \left(a\right)\frac{1}{\sqrt{3}}\\ \left(ii\right)cot{30}^{\circ }& \left(b\right)\sqrt{3}\\ \left(iii\right)cosec{30}^{\circ }& \left(c\right)2\\ \left(iv\right)sec{30}^{\circ }& \left(d\right)\frac{2}{\sqrt{3}}\end{array}$

1. (i) -a , (ii) - b , (iii) - d , (iv) - c
2. (i) -b , (ii) - a , (iii) - d , (iv) - c
3. (i) -b , (ii) - b , (iii) - c , (iv) - d
4. (i) -a , (ii) - b , (iii) - c , (iv) - d

Q. $ABC$ is a triangular park with $AB=AC=100$ metres. A vertical tower is sitiuated at the mid-point of $BC$. If the angle of elevation of the top of the tower at $A$ and $B$ are ${\cot }^{-1}\left(3\sqrt{2}\right)$ and ${\text{cosec}}^{-1}\left(2\sqrt{2}\right)$ respectively, then the height of the tower (in metres) is :

1. $\frac{100}{3\sqrt{3}}$
2. $25$
3. $20$
4. $10\sqrt{5}$

Q.

How many of the following statements are correct?

Q. How do you find cos if $sin=5/13$ ?

Q. Prove that $-2\sin \frac{A}{2}=-\sqrt{1+\sin A}-\sqrt{1-\sin A}$ ?

Q. In an equilateral triangle $ABC$ , match the following Trigonometric ratios to the values
$\begin{array}{cc}Trigonometricratios& Values\\ \left(i\right)tan{60}^{\circ }& \left(p\right)\frac{1}{\sqrt{3}}\\ \left(ii\right)cot{30}^{\circ }& \left(q\right)\frac{2}{\sqrt{3}}\\ \left(iii\right)cosec{30}^{\circ }& \left(r\right)2\\ \left(iv\right)sec{30}^{\circ }& \left(s\right)\sqrt{3}\end{array}$

1. (i) -b , (ii) - b , (iii) - c , (iv) - d
2. (i) -a , (ii) - b , (iii) - c , (iv) - d
3. (i) -a , (ii) - b , (iii) - d , (iv) - c
4. (i) -b , (ii) - a , (iii) - d , (iv) - c

Q. In triangle ABC, right angled at B, Find the expressions of cos A , sin C, tan A and cot C.

1. AB/AC , BC/AC, BC/AB , AB/BC

2. BC/AC , AB/AC, BC/AB , BC/AB

3. AB/AC , AB/AC, BC/AB , BC/AB

4. BC/AC , BC/AC, AB/BC , BC/AB

Q. The value of the expression $\frac{(1+\cos 2A)(1-\sec 2A)}{\cot A(1+\tan A)(1-\cot A)}$, when $A={30}^{\circ }$ is

Q. The value of the expression $\frac{(1+\tan \frac{\pi }{6})(1-\cot \frac{\pi }{6})}{(1+\cos \frac{\pi }{3})(1-\sec \frac{\pi }{3})}$ is

1. $\frac{4}{3\sqrt{3}}$
2. $\sqrt{3}$
3. $\frac{4}{\sqrt{3}}$
4. $3\sqrt{3}$

Q. From the top of a building $21\text{m}$ high, the angle of elevation and depression of the top and the foot of another buillding are ${30}^{\circ }$ and ${45}^{\circ }$ respectively. The height of the second building is

1. $21+7\sqrt{3}$
2. $21\sqrt{3}$
3. $21\sqrt{3}+7$
4. $7\sqrt{3}$