Trigonometry Questions and Answers with Solutions

Q1 :

The value of $theta left(0 leq theta leq frac{pi}{2} right)$ satisfying the equation

$text{sin}^2 theta - 2 text{ cos} theta + frac{1}{4} = 0 text{ , is}$

A

$frac{pi}{2}$

B

$frac{pi}{3}$

C

$frac{pi}{4}$

D

$frac{pi}{6}$

Right Answer: B

$frac{pi}{3}$

Q2 :

The expression for sin $theta$ in terms of sec $θ$

A

$frac{1}{sqrt{text{ sec}^2 theta - 1 }}$

B

$frac{text{ sec}^2 theta}{sqrt{text{ sec}^2 theta - 1 }}$

C

$frac{sqrt{text{ sec}^2 theta - 1 }}{text{ sec} theta}$

D

$sqrt{text{ sec}^2 theta - 1 }$

Right Answer: C

$frac{sqrt{text{ sec}^2 theta - 1 }}{text{ sec} theta}$

Q3 :

The value of $frac{text{sin } 45^circ + text{ sin } 105^circ + text{ cos } 105^circ text{ cos } 45^circ} { text{sin } 75^circ + text{ cos } 75^circ }$ is equal to

A

$frac{1}{sqrt6}$

B

$frac{1}{sqrt3}$

C

$frac{1}{2sqrt3}$

D

$frac{1}{3sqrt2}$

Right Answer: A

$frac{1}{sqrt6}$

Q4 :

If $text{ tan}x + text{cot}x = 3, text{then sec}^2x + text{cosec}^2x$ is equal to

A

3

B

9

C

12

D

15

Q5 :

Two boys are on opposite sides of a tower of 100 meter height. If they measure the elevation of the top of the tower as $30^circ$ and $45^circ$ respectively, the distance between the tower is 200 metres, then the distance between the boys is

A

$100sqrt3 text{ metres}$

B

$100(sqrt3 +1) text{ metres}$

C

$100(sqrt3 -1) text{ metres}$

D

$100sqrt3 -1 text{ metres}$

Right Answer: C

$100(sqrt3 -1) text{ metres}$

Q6 :

For $lambda ne 0$ , the angle between the lines given by the equation $lambda y^2 + (1 - lambda^2) ; xy - lambda x^2 = 0$ is

A

$30degree$

B

$45degree$

C

$60degree$

D

$90degree$

Right Answer: D

$90degree$

Q7 :

If $text{cos } alpha + text{sec } alpha = 2$ , then the value of $text{cos}^8 alpha + text{sec}^8 alpha$ is equal to

A

$2$

B

$2^2$

C

$2^4$

D

$2^8$

Right Answer: A

$2$

Q8 :

The numerical value of the expression

$dfrac{text{sin}9degree}{text{sin}48degree} - dfrac{text{cos}81degree}{text{cos}42degree}$ $frac{text{sin}9degree}{text{sin}48degree} - frac{text{cos}81degree}{text{cos}42degree}$ is

A

1

B

1/2

C

0

D

-1

Q9 :

An angle is divided into two parts $alpha$ and $beta$ in such $alpha$ way that $tan alpha = frac{1}{2}$ and $tan beta = 2$ . The measure of the angle is

A

$frac{2pi}{3}$

B

$frac{pi}{2}$

C

$pi$

D

$frac{3pi}{4}$

Right Answer: B

$frac{pi}{2}$

Q10 :

If $alpha + beta = 90degree$ , then $text{cosec}^2alpha + text{cosec}^2beta$ is equal to

A

$text{cosec}^2alpha + text{cosec}^2beta$

B

$text{sin}^2alpha + text{sin}^2beta$

C

$text{tan}^2alpha + text{tan}^2beta$

D

$text{sec}^2alpha + text{sec}^2beta$

Right Answer: A

$text{cosec}^2alpha + text{cosec}^2beta$

Q11 :

If $sin 2theta =cos 3theta$ and $theta$ is an acute angle, then $theta$ is equal to

A

$18degree$

B

$27degree$

C

$36degree$

D

$45degree$

Right Answer: A

$18degree$

Q12 :

If sec $11theta = text{ cosec } 7theta (0degree lt theta lt 20degree)$ , then the value of $theta$ is

A

$5degree$

B

$10degree$

C

$15degree$

D

$18degree$

Right Answer: A

$5degree$

Q13 :

The maximum value of $sin theta cdot cos theta$ is

A

1

B

$frac{1}{2}$

C

$frac{1}{sqrt{2}}$

D

$frac{sqrt{3}}{2}$

Right Answer: B

$frac{1}{2}$

Q14 :

The value of $sin^3 (15degree) - cos^3 (15degree)$ is

A

$frac{3}{4}(sin 15degree - cos 15degree )$

B

$frac{5}{8sqrt{2}}$

C

$-frac{5}{8sqrt{2}}$

D

$-frac{5}{4sqrt{2}}$

Right Answer: D

$-frac{5}{4sqrt{2}}$

Q15 :

The value of $frac{3 tan 20degree - tan^3 20 degree}{1 - 3tan^2 20}$ is equal to

A

$frac{1}{sqrt{3}}$

B

1

C

$sqrt{3}$

D

$frac{2}{sqrt{3}}$

Right Answer: C

$sqrt{3}$

Q16 :

If $2 cos^2 theta + 11 sin theta - 7 = 0$ , then the value of $sin theta$ is equal to

A

$frac{-1}{2}$

B

$frac{1}{2}$

C

5

D

$frac{1}{sqrt{2}}$

Right Answer: B

$frac{1}{2}$

Q17 :

The angle between the hour and minute hands of a clock at 02 : 15 hour is

A

$15degree$

B

$7frac{1}{2}degree$

C

$22frac{1}{2}degree$

D

$30degree$

Right Answer: C

$22frac{1}{2}degree$

Q18 :

An aeroplane at a height of 3000 m, passes vertically above another aeroplane at an instant. If the angles of elevation of the two aeroplanes from some point on the ground are $60degree$ and $45degree$ , respectively, then the vertical distance between the two planes is:

A

$1000 Big(sqrt{3 - 1}Big) text{ m }$

B

$1000 sqrt{3} text{ m }$

C

$1000 Big(3 - sqrt{3}Big) text{ m }$

D

$1000 sqrt{3} text{ m }$

Right Answer: C

$1000 Big(3 - sqrt{3}Big) text{ m }$

Q19 :

If $tan theta + sec theta = 2, 0 le theta le frac{pi}{2}$ ; then the value of $tantheta$ is equal to :

A

$frac{3}{4}$

B

$frac{5}{4}$

C

$frac{3}{2}$

D

$frac{5}{2}$

Right Answer: A

$frac{3}{4}$

Q20 :

The value of $(cos^2 25degree + cos^2 65degree)$ is:

A

0

B

$sin^2 40degree$

C

$cos^2 40degree$

D

1