Trigonometric Ratios of Common Angles

Q.

A man is observing, from the top of a tower, a boat speeding towards the tower from a certain point $A$, with uniform speed.

At that point, the angle of depression of the boat with the man’s eye is $30°$ (Ignore the man’s height).

After sailing for $20$ seconds towards the base of the tower (which is at the level of water), the boat has reached point $B$, where the angle of depression is$45°$.

Then the time taken (in seconds) by the boat from $B$ to reach the base of the tower is :

1. $10(\sqrt{3}-1)$

2. $10\sqrt{3}$

3. $10$

4. $10(\sqrt{3}+1)$

Q.

From the top of a tower, the angle of depression of a point on the ground is $60°$. If the distance of this point from the tower is $\frac{1}{(\sqrt{3}+1)}m$, then the height of the tower is

1. $\frac{\left(4\sqrt{3}\right)}{2}$m

2. $\frac{(\sqrt{3}+3)}{2}$m

3. $\frac{(3–\sqrt{3})}{2}$m

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

5. $\sqrt{3}+1$m

Q.

In a triangle ABC, a = 3, b = 5, c = 7. Find the angle opposite to C.

1. ${60}^{\circ }$

2. ${90}^{\circ }$

3. ${120}^{\circ }$

4. ${150}^{\circ }$

Q.

prove that $\tan 70°=\tan 20°+2\tan 50°$

Q. If in a triangle a = $\sqrt{3}+1,b=\sqrt{3}-1,C={60}^{\circ }$ then the value of A is ?

1. ${75}^{\circ }$
2. ${120}^{\circ }$
3. ${90}^{\circ }$
4. ${105}^{\circ }$

Q. A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank is ${45}^{\circ }.$ When he moves $20\text{m}$ away from the bank, he finds the angle of elevation to be ${30}^{\circ }.$ The height of the tree is

1. $10(\sqrt{3}+1)\text{m}$
2. $15\sqrt{3}\text{m}$
3. $20(\sqrt{3}+1)\text{m}$
4. $10(\sqrt{3}-1)\text{m}$

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. The angle of elevation of the top of a vertical tower from a point A, due east of it is ${45}^{\circ }$. The angle of elevation of the top of the same tower from a point B, due south of A is ${30}^{\circ }$. If the distance between A and B is $54\sqrt{2}$ m, then the height of the tower (in metres), is

1. $108$
2. $54\sqrt{3}$
3. $36\sqrt{3}$
4. $54$

Q.

If tan(A - B)=1, sec (A + B)= $\frac{2}{\sqrt{3}}$, then the smallest positive value of B is

1. $\frac{25}{24}\pi$

2. $\frac{19}{24}\pi$

3. $\frac{13}{24}\pi$

4. $\frac{11}{24}\pi$

Q.

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

1. 0

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

3. 1

4. $\infty$

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. If $\cos (\alpha -\beta )=1$ and $\cos (\alpha +\beta )=\frac{1}{2},$ where $\alpha ,\beta \in (-\pi ,\pi ),$ then the number of ordered pairs $(\alpha ,\beta )$ satisfying both the equations is

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

Q.

If tan(A - B)=1, sec (A + B)= $\frac{2}{\sqrt{3}}$, then the smallest positive value of B is

1. $\frac{25}{24}\pi$

2. $\frac{19}{24}\pi$

3. $\frac{13}{24}\pi$

4. $\frac{11}{24}\pi$

Q. The value of $\cos 1^{\circ }\cdot \cos 2^{\circ }\cdot \cos 3^{\circ }\cdots \cos {180}^{\circ }$ is

1. $1$
2. $\sqrt{2}$
3. $0$
4. $-1$