## Kerala Plus One Maths Chapter Wise Questions and Answers Chapter 3 Trigonometric Functions

**Short Answer** **Type Questions **

**(Score 3)**

Question 1.

In a triangle ABC, a = 4, c = 12, and c = 60°, then find the value of sin A.

Answer:

Question 2.

Let angle x lie in III ^{rd} quadrant

i. Complete the following table:

Answer:

Question 3.

a. 135°=……………… radian

b. Prove that 1 + cos 2x = 2cos^{2}x

Answer:

a. degree measure = Radian measure × 135° =3 radian .

b.

sin^{2}x + cos^{2}x = 1

cos2x = cos^{2}x – sin^{2}x

1+cos2x = sin^{2}x + cos^{2}x + cos^{2}x – sin^{2}x

= 2cos^{2}x

Question 4.

The minute hand of a clock is 18cms long

i. Find angle described by the minute hand in 10 minutes in radians

ii. How many centimetres does the extremity of the minute hand move in 10 minutes?

Answer:

i. The angle described by the minute hand of a clock in 60 minutes =2π radian

∴ The angle described by the minute hand in 10 minutes

ii. r=18,l=rθ=18× = 6π cm

Question 5.

i. Find the radian measure corresponding to 75°.

ii. The arcs of same length in two circles subtends 60° and 75° at the centre. Find the ratio of their radii.

Answer:

Question 6.

Match the following :

Answer:

Question 7.

If tan A= and tan B= ,show that 2A +B=.

Answer:

Question 8.

a. Show that sin 78° – sin 18° +cos 132° = 0

b. If cosθ = and θ is an angle in the fourth quadrant, find the value of

Answer:

a. LHS = 2cos 48° sin 30°+ cos (180° – 48°)

= cos 48° – cos 48° = 0 = RHS -24 24

b. tanθ = , sin

∴ answer is 25

Question 9.

Prove that

Answer:

Question 10.

i. prove that

ii. Hence deduce the value of sin 150°

Answer:

Question 11.

a. Convert radian measures into degrees measures.

b. Prove that :

Answer:

.

Question 12.

a. If cosα + cosβ = and sin α +sin β = ,

show that

b. Prove that

Answer:

Question 13.

Prove that the equation 2sinx+cosx = 3 has no solution.

Answer:

2 sin x + cos x – 3

2 sinx = 3 – cosx

(2sinx)^{2} = (3-cosx)^{2 }4 sin^{2}x = 9-6 cos x + cos^{2} x

4 (1 -cosx) = 9-6 cos x + cos^{2}x

5 cos^{2}x – 6 cos x + 5 = 0

Range of cosine is [-1,1]

The above equation does not satisfied the condition. So it has no solution.

**Short Answer** **Type Questions**

**(Score 4)**

Question 1.

a. Find the exact value of

b. If sinθ=nsin(θ+2α), show that tan(θ+α)=

Answer:

Question 2.

i. Simplify

ii. Find the value of

iii. Hence show that tan 10° tan 20° tan 70° tan 80° = 1.

Answer:

Question 3.

a. Prove that tan 70° = tan 20° + 2 tan 50°

b. Prove that

Answer:

Question 4.

i. Find all the solutions of the equation sin 3x = 0

ii. Solve cos 2x = -1/2

iii. Also And the general solution of the equation sin x + sin 3x + sin 5x = 0

Answer:

Question 5.

Prove that

b. prove that sin θ≠-

Answer:

Question 6.

Observe the clock, time in the clock is 8.20.

i. What are the angles traced by the hour hand in 12 hours and the minute hand in 60 minutes.

ii. Find the angle between the hour hand and the minute hand at the time shown in the above clock in degree measure.

iii. Find the above angle in radian measure.

Answer:

i. Both are 360°

ii. Angle traced by the hour hand in 8 hrs. 20mts = 250°

Angle traced by the minute hand in 20 mts = 120°

Required angle =130°

iii.

Question 7.

Show that

Answer:

Question 8.

a.Which of the following is not possible

i. Sin x=

ii. cos x =

iii.

iv. tanx=8

b. Find the value of sin 15°.

c. Hence write the value of cos 75°.

Answer:

**Long Answer Type questions **

**(Score 6)**

Question 1.

a. If sinx = cos x, x ∈ [0,π] then x is

i. 0

ii.

iii.

iv. π

b. Write the following in ascending order of its values.

c. sin 100°, sin 0°, sin 50°, sin 200°. c. Solve : sin2x – sin 4x + sin 6x = 0

Ans:

Question 2.

i. Complete the following table:

ii. Show that 2sin^{2} + sin^{2} =1

iii. Find the general solution is sinx+cos x =1

Answer:

Question 3.

Consider the trigonometric expression

i. Of the above expressions which one is tan A and which one is cot A?

ii. Evaluate tan 22°30′ and cot 22°30′.

iii.

Answer:

Question 4.

i. Express sin3A and cos2A in terms of sine function

ii. Convert sin5A + sinA into product form and show that sin 5A = 5sin A- 20sin^{3}A + 16sin^{5}A

iii. If A=36°, deduce from the above result

Answer:

i. sin 3A = 3sin A – 4sin^{3}A

cos 2A = 1 – 2 sin^{2}A

ii. sin 5A+ sin A= 2sin 3A cos 2A

sin 5A+ sinA= 2(3sinA-4sin^{3}A)(1- 2sin^{2}A)

= 2 (3 sin A -10 sin^{3}A + 8 sin^{5}A)

iii. sin 5A= 5sin A – 20 sin^{3}A+ 16 sin^{5}A

Put A= 36°, Then

sin 180 = 5 sin 36°-20 sin^{3}36°+ 16 sin^{5}36°

0 = sin 36° (5 – 20 sin^{2} 36°+ 16 sin^{4}36°)

16 sin^{4} 36° -20 sin^{2}36° + 5 = 0

By using quadratic formula

Question 5.

i.tan =………………

Answer:

Question 6.

i. tan(A+B) =……………..

Answer:

Question 7.

Prove that

Answer:

.

Question 8.

Sketch the graphs of the functions y = sinx and y = sin 2x on the same axes.

b. Solve

Answer:

Question 9.

If A + B + C = 180°, prove that cos^{2}A+ cos^{2}B + cos^{2}C = 1- 2cosA cosBcos C.

Answer:

= 2 + 2 cos (A + B). cos (A – B)+2cos^{2}C

= 2-2 cos C cos (180 -C)tcos (A-B) + 2cos^{2}C

= 2-2 cosC cos (A – B) + 2 cos^{2}C = 2-2 cosC [cos(A – B) – cos C]

= 2-2 cosC[cos(A-B) – cos(180 – (A + B))]

= 2-2 cosC [ cos(A – B)+ cos (A + B)]

= 2 – 2cosC.2cosAcosB = 2 – 4cosA cosB cosC

∴ S = 1-2cosA cosB cosC or cos^{2}A+cos^{2}B+ cos^{2}C

= 1-2cosA cosB cosC

**NCERT Questions and Answers**

Question 1.

Convert 6 rad into degree measure.

Answer:

Question 2.

Find the radian measure corresponding to the following degree measures,

a. 25°

b. -47° 30’

Answer:

Question 3.

If cosθ=- and 0 lies in III quadrant, then Find other five trigonomentric function.

Answer:

Question 4.

Answer:

Question 5.

In a triangle ABC, if a cos A = b cos B, then show that the triangle is either isosceles or right angled.

Answer:

Question 6.

A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

Answer:

Number of revolutions in 60 seconds = 360

Number of revolutions in 1 sec

In one revolution, the wheel turns 2 n radian.

In 6 revolutions, the wheel turns

2 π x 6 = 12 K radian.

Question 7.

In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of the minor arc of the chord.

Answer: