Values of Trigonometric Ratios

Q.

The shadow of a tower standing on a level plane is found to be $50\mathrm{m}$longer when Sun’s elevation is $30°$ than when it is$60°.$ Find the height of the tower.

Q.

Considering only the principal values, if $\mathrm{ta}n\left({\cos }^{-1}x\right)=\sin \left[co{t}^{-1}\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)\right]$ then $x=$

1. $\frac{1}{\sqrt{5}}$

2. $\frac{2}{\sqrt{5}}$

3. $\frac{3}{\sqrt{5}}$

4. $\frac{\sqrt{5}}{3}$

Q.

How do you evaluate $\tan (-30°)$?

Q. The value of $cose{c}^2{30}^{\circ }si{n}^2{45}^{\circ }-se{c}^2{60}^{\circ }$ is

1. 1
2. 2
3. -2
4. $2$

Q.

In $△$PQR, PQ = 12 cm and PR = 13 cm. $\angle Q={90}^{\circ }$. Find tan P - cot R

1. $\frac{119}{60}$

2. $-\frac{119}{60}$

3. $1$

4. $0$

Q.

Say true or False: The value of tanA is always less than 1.

1. True

2. False

Q. Question 11
The value of the expression
$(\frac{si{n}^2{22}^{\circ }+si{n}^2{68}^{\circ }}{co{s}^2{22}^{\circ }+co{s}^2{68}^{\circ }}+si{n}^2{63}^{\circ }+cos{63}^{\circ }{27}^{\circ })$ is

(A) 3
(B) 2
(C) 1
(D) 0

Q.

The value $co{t}^2{30}^{\circ }-2co{s}^2{60}^{\circ }-\frac{3}{4}se{c}^2{45}^{\circ }-4si{n}^2{30}^{\circ }$ is

1. 0

2. 1

3. -1

4. 2

Q.

The value of
$\frac{co{t}^2{30}^{\circ }+8si{n}^2{45}^{\circ }+\frac{3}{2}se{c}^2{30}^{\circ }+2co{s}^2{90}^{\circ }}{2sec{60}^{\circ }+3cosec{30}^{\circ }-\frac{7}{3}ta{n}^2{60}^{\circ }}$ is :

1. 4

2. 2

3. 3

4. 1

Q.

Say True or False:

$sec\theta =cosec(90-\theta )$ for acute values of $\theta$

1. True

2. False

Q.

In a right angled triangle, if AB=3 cm, BC=4 cm then what is the value of cos A?

1. $\frac{3}{5}$

2. $\frac{4}{5}$

3. $\frac{3}{4}$

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

Q.

$cos{30}^{\circ }cos{60}^{\circ }-sin{30}^{\circ }sin{60}^{\circ }$= __________________

1. 2

2. -1

3. 0

4. 1

Q. Question 2 (iv)
Choose the correct option and justify your choice.
(iv) $\frac{2tan30^0}{1-ta{n}^2{30}^0}=$
(A) $cos{60}^{\circ }$
(B) $sin{60}^{\circ }$
(C) $tan{60}^{\circ }$
(D) $sin{30}^{\circ }$

Q.

Derive the formula used to find the area of segment of circle. [4 MARKS]

Q.

If x is an acute angle of a right triangle and sin x = 3 / 7, find the exact value of the trigonometric function cos x.

1. 2 $\sqrt{(}10)/3$

2. 2 $\sqrt{(}10)/7$

3. 3 $\sqrt{(}10)/3$

4. 5 $\sqrt{(}10)/3$

Q.

If $cot3x=sin{45}^{\circ }cos{45}^{\circ }+cos{60}^{\circ }$ and $0<x\le {90}^{\circ }$ , then $x$= __________________

1. ${15}^{\circ }$

2. ${30}^{\circ }$

3. ${45}^{\circ }$

4. ${60}^{\circ }$

Q.

In a right angled triangle, as shown below, if AB = 3 cm, BC = 4 cm, then what is the value of cos A?

1. 3/4

2. 2/3

3. 3/5

4. 4/5

Q. In a rectangle $ABCD$, $AB=20cm$, $\angle BAC={60}^{\circ }$, Then side $BC=$

1. 
2. 
3. 
4. 

Q. Question 4
Given 15 cot A = 8, find sin A and sec A.

Q.

The value of $e^{{\log }_{10}\tan 1^{\circ }+{\log }_{10}\tan 2^{\circ }+{\log }_{10}\tan 3^{\circ }+.......+{\log }_{10}\tan {89}^{\circ }}$is

1. $0$

2. $e$

3. $\frac{1}{3}$

4. None of these

Q. Question 11 (v)
State whether the following are true or false. Justify your answer.
(v) sin $\theta$ = $\frac{4}{3}$ for some angle $\theta$.

Q. ABCD is a rhombus with each side of length 20 cm. Angle ABC and ADC measure ${60}^{\circ }$ each. The length of the diagonal AC is :

1. 10 cm
2. 20 cm
3. 15 cm
4. None of these

Q.

The value of
$\frac{2tan{30}^{\circ }}{1-ta{n}^2{30}^{\circ }}$ is _______.

1. $tan{30}^{\circ }$

2. $tan{60}^{\circ }$

3. $tan{0}^{\circ }$

4. $tan{90}^{\circ }$

Q.

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

1. √2, √2

2. 1/√2, 1/√2

3. 1, 1

4. 1/√3, 1/√3

Q.

In the given right angle triangle, if Sin$\theta$ =$\frac{3}{5}$

The value of 3tan$\alpha$ is :

1. 3

2. 4

3. 5

4. 6

Q.

If TanA = $\frac{12}{5}$, then the value of (sinA+cosA) $\times$ secA is:

1. 13/5

2. 17/5

3. 14/5

4. 1

Q. Question 5 (v)
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(v) $\frac{(cosA-sinA+1)}{(cosA+sinA-1)}=cosecA+cotA$, using the identity $cose{c}^{2}A=1+co{t}^{2}A$.

Q.

Given that sin $A=\frac{1}{2}$ and $cosB=\frac{1}{\sqrt{2}}$, then the value of (A + B) is:

1. 30 degrees

2. 45 degrees

3. 75 degrees

4. 15 degrees

Q. Question 4 (iii)
State whether the following are true or false. Justify your answer.
(iii) The value of cos $\theta$ increases as $\theta$ increases.

Q. Find the value of $si{n}^2\theta +co{s}^2\theta at\theta ={30}^{\circ }\&{60}^{\circ }$ respectively.

1. $\frac{1}{4},\frac{3}{4}$
2. $1,1$
3. $0,\frac{3}{4}$
4. $\frac{3}{4},\frac{1}{4}$