1 Marks Questions

1. Find the radian measure corresponding to 5° 37" 30""

Ans.

2. Find the degree measure corresponding to

Ans. 39°22"30""

3. Find the length of an arc of a circle of radius 5 cm subtending a central angle measuring 15°

Ans.

4. Find the value of

Ans.

5. Find the value of sin(–1125°)

Ans.

6. Find the value of tan 15°

Ans.

7. If sin A = and < A < , find cos A

Ans.

8. If tan A = and tan B = then find the value of A + B.

Ans.

9. Express sin 12θ + sin 4θ as the product of sines and cosines.

Ans. 2 sin8θ cos4θ

10. Express 2 cos4x sin2x as an algebraic sum of sines or cosines.

Ans. sin 6x – sin2x

11. Write the range of cos θ

Ans. [–1,1]

12. What is domain of sec θ?

Ans.

13.Find the principal solutions of cotx = 3 

Ans.

14. Write the general solution of cos θ = 0

Ans.

15. If sinx = and 0 < x < find the value of cos 2x

Ans.

16. If cosx = and x lies in quadrant III, find the value of sin

Ans.

17.Convert into radian measures. – 47⁰ 30ʹ

Ans. - 47⁰ 30ʹ = -

18.Evaluate tan 75⁰.

Ans. tan 75 = tan (45 + 30)

19.Prove that Sin (40 + θ). Cos (10 + θ) – Cos (40 + θ). Sin (10 + θ) =

Ans. L. H. S = Sin (40 + θ). Cos (10 + θ) – Cos (40 + θ). Sin (10 +θ)

= Sin

20.Find the principal solution of the eq. Sin x =

Ans.Sin x =

21.Prove that

Ans. L. H. S = Cos

22.Convert into radian measures. -37⁰ 30’

Ans. - 37⁰ 30’ = -

23.Prove Sin (n+1) x Sin (n+2) x + Cos (n+1) x. Cos (n+2) x = Cos x

Ans.L.H.S = Cos (n+1) x Cos (n+2) x + Sin (n+1) x Sin (n+2) x

=Cos

=Cos x

24.Find the value of Sin

Ans. Sin = Sin

= sin

= Sin

=

25.Find the principal solution of the eq. tan x =

Ans. tan x =

26.Convert into radian measures.

Ans. 5⁰ 37¹ 30¹¹ = 5⁰ +

27.Prove Cos 70⁰. Cos 10⁰ + Sin 70⁰. Sin 10⁰ =

Ans. L. H. S. = Cos (70 – 10) = Cos 60 =

28.Evaluate 2 Sin

Ans.2 Sin

29.Find the solution of Sin x =

Ans.Sin x =

30.Prove that

Ans.L. H. S = tan 36⁰

31.Find the value of tan

Ans.

32.Prove Cos 4x = 1 – 8 Sin² x. Cos²x

Ans. L. H. S = Cos 4x

33.Prove

Ans.L. H. S =

34.Prove that tan 56⁰ =

Ans.L. H. S = tan 56⁰

= tan (45⁰ + 11⁰)

35.Prove that Cos 105⁰ + Cos 15⁰ = Sin 75⁰ – Sin 15⁰

Ans.L. H. S = Cos 105⁰ + Cos 15⁰

36.Find the value of Cos (- 1710⁰).

Ans. Cos (-1710⁰) = Cos (1800-90)[Cos (-θ) = Cos θ

= Cos [5 360 +90]

= Cos = 0

37.A wheel makes 360 revolutions in 1 minute. Through how many radians does it turn in 1 second.

Ans.N. of revolutions made in 60 sec. = 360

N. of revolutions made in 1 sec =

Angle moved in 6 revolutions = 2 π 6 = 12 π

38.Prove Sin² 6x – Sin² 4x = Sin2 x. Sin 10 x.

Ans.L. H. S = Sin² 6x – Sin² 4x

= Sin (6x + 4x). Sin (6x – 4x)

= Sin 10x . Sin 2x

39.Prove that

Ans.L. H. S = tan (69 + 66)

= tan (135)

= tan (90 + 45)

= - tan 45

= -1

40.Prove that

Ans.L. H. S

= tan

4 Marks Questions

Prove the following Identities

1.The minute hand of a watch is 1.5 cm long. How far does it tip move in 40 minute?

Ans. r = 1.5 cm

Angle made in 60 mint = 360⁰

Angle made in 1 min = = 60⁰

Angle made in 40 mint = 6 40

= 240⁰

Θ =

2. Show that tan 3x. tan 2x. tan x = tan 3x – tan 2x – tan x

Ans.Let 3x = 2x + x

tan 3x = tan (2x + x)

3.Find the value of tan .

Ans.Let x =

4.Prove that

Ans.L.H.S =

5.If in two circles, arcs of the same length subtend angles 60⁰ and 75⁰ at the centre find the ratio of their radii.

Ans.

(1)÷ ( 2)

6.Prove that Cos 6x= 32 Cos⁶x – 48 Cos⁴ x + 18 Cos² x-1

Ans.L.H.S. = Cos 6x

=

7.Solve Sin2x-Sin4x+Sin6x=o

Ans.

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

Ans.

Θ = 60⁰

s

9.Prove that tan 4x =

Ans. L. H. S = tan 4x

10.Prove that (Cos x + Cos y)² + (Sin x – Sin y)² = 4 Cos²

Ans. L. H. S = (Cos x + Cos y)² + (Sin x – Sin y)²

11.If Cot x = - x lies in second quadrant find the values of other five trigonometric functions.

Ans.Cot x =

12.Prove that

Ans.L. H. S =

13.Prove that Sin x + Sin 3x + Sin 5x + Sin 7x = 4 Cos x. Cos 2x. Sin 4x

Ans.L. H. S. = Sin x + Sin 3x + Sin 5x + Sin 7x

= Sin x + Sin 7x + Sin 3x + Sin 5x

14.Find the angle between the minute hand and hour hand of a clock when the time is 7. 20.

Ans. Angle made by mint hand in 15 mint= 15 6 = 90⁰

Angle made by hour hand in 1 hr = 30⁰

in 60 minute = =

in 20 minute = =

Angle made = 90 + 10 = 100⁰

15.Show that

Ans.L H. S =

16.Prove that Cot 4x (Sin 5x + Sin 3x) = Cot x (Sin 5x – Sin 3x)

Ans.L. H. S = Cot 4x (Sin 5x + Sin 3x)

R. H. S = Cot x (Sin 5x – Sin 3x)

6 Marks Questions

1. Find the general solution of sin2x + sin4x + sin6x = 0

Ans.

2. Find the general solution of cos θ cos2 θ cos3θ =

Ans.

3.If Sin α + Sin β = a and Cos α + Cos β = b show that Cos (α + β) =

Ans.

4. Prove that Cos α + Cos β + Cos γ + Cos (α + β + γ)

Ans.L. H. S.

5.Prove that Sin3x +Sin2x-Sin2x=4Sinx.Cos. Cos

Ans.

6.Prove that 2Cos .Cos + Cos + Cos=0

Ans.L.H.S.

7. Find the value of tan (α + β) Given that

Ans.

8.Prove that

Ans.L. H. S=

9.Prove that

Ans.L. H. S =

10..Prove that Cos 2x. Cos -

Ans.L. H. S =

11.Prove that Cos 20⁰. Cos 40⁰. Cos 60⁰ Cos 80⁰ =

Ans.L. H. S = Cos 20⁰. Cos 40⁰. Cos 60⁰. Cos 80⁰.

= Cos 60. Cos 20⁰. Cos 40⁰. Cos 80.

12.If tan x =

Ans.π< x <

Cos x is – tive