MP Board Class 11th Maths Important Questions Chapter 3 Trigonometric Functions

MP Board Class 11th Maths Important Questions Chapter 3 Trigonometric Functions

MP Board Class 11th Maths Important Questions Chapter 3 Trigonometric Functions

Trigonometric Functions Objective Type Questions

(A) Choose the correct option :
Question 1.
The value of 1 + cosθ is :
(a) 2sin² θ
(b) sin²θ/2
(c) 2cos² θ
(d) cos² θ
Answer:
(c) 2cos² θ

Question 2.
The value of
(a) cot 56°
(b) cot 34°
(c) tan 34°
(d) tan 56°
Answer:
(d) tan 56°

Question 3.
The value of sin 18° is :

Answer:

Question 4.
The value of cos 1° cos 2° cos 3° ………… cos 179° is :
(a) 0
(b) 1
(c) – 1
(d) None of these
Answer:
(a) 0

Question 5.
The value of 3π/2 radian in degree.
(a) 120°
(b) 170°
(c) 220°
(d) 270°
Answer:
(d) 270°

Question 6.
The value of cos² (60 + α) + cos² (60 – α) + cos² α = ……………

Question 7.
Amplitude of the function f(x) = 2 sin x is :
(a) π
(b) 2π
(c) 1
(d) 2
Answer:
(d) 2

Question 8.
Period of the function f(x) = tan x is :
(a) π
(b) 2
(c) π/2
(d) – π
Answer:
(a) π

Question 9.
Solution of the equation 4 sin²θ = 1 is :

Question 10.
Maximum value of 3cosθ + 4sinθ is :
(a) 3
(b) 4
(c) 5
(d) 7
Answer:
(c) 5

Question 11.

Question 12.
The value of tan15° + cot15° is :
(a) 1
(b) 3
(c) 2
(d) 4
Answer:
(d) 4

Question 13.
The value of sin50° + sin70° + sin10° is :
(a) 0
(b) 1
(c) – 1
(d) None of these
Answer:
(a) 0

Question 14.
One value of θ which satisfy the equation cosθ +\/3sinθ = 2 is :

Question 15.
The general value of θ satisfying the equation cosθ = , tanθ = 1 is :

(B) Match the following :

Answer:

1. (d)
2. (c)
3. (a)
4. (b)
5. (g)
6. (j)
7. (e)
8. (i)
9. (h)
10. (f)

(C) Fill in the blanks :

1. The value of cos 18° = ……………….
2. The value of sin 75° = ……………….
3. If tan A = 5/6 and tan B = 1/3, then the value of A + B = ……………….
4. 2π radian is equal to ………………. right angle.
5. If sinθ + cosθ = 1, then the value of sinθ.cosθ = ……………….
6. The value of  = ……………….
7. Solution of the equation cos2θ = cos²θ is ……………….
8. Amplitude of 3cosx is ……………….
9. The value of cot 22½° is ……………….
10. If tan θ tan 2θ = 1, then the value of θ is ……………….
11. The value of sin(A + B).sin(A – B) = ……………….
12. The length of the arc of a circle of radian 6 cm substending an angle of 30° at the centre of the circle is ……………….

Answer:

(D) Write true / false :

Answer:

1. False
2. False
3. True
4. True
5. False
6. False
7. False
8. True
9. False
10. True

Trigonometric Functions Short Answer Type Questions

Question 1.
A wheel makes 360 revolutions in 1 min. then how many radians measure of an angle does it turn in 1 second? (NCERT)
Solution:
In 60 seconds number of revolutions of wheel = 360
∴ In 1 second number of revolutions of wheel = 360/60
∴ In one revolution angle made = 360° = 2π
∴ In 6 revolutions angle made = 2π x 6 = 12π radian

Question 2.
Find the degree measure of an angle substended at the centre of a circle of radius 100 cm by an arc of length 22 cm.
Solution:

Question 3.
In a circle of diameter 40 cm, the length of chord is 20 cm, then find the length of minor arc of the chord.
Solution:

Question 4.
If in two circles, arc of the same length substend angles of 60° and 75° at the centre, then find the ratio of their radii. (NCERT)
Solution:

Question 5.
Find value : (i) sin75°, (ii) tan15°
Solution:

Question 6.
Prove that:

Solution:

Question 7.
Prove that:
cos 27°. tan 27°. tan 63°. cosec 63° = 1. (NCERT)
Solution:
L.H.S. = cos 27°. tan 27°.tan 63°. cosec 63°
= cos 27°. tan 27° tan(90° – 27°). cosec (90° – 27°)
= cos 27°. tan 27° cot 27°. sec 27°

= 1 = R.H.S.

Question 8.
Find the general solution of the following equations :

Question 9.
Solve the following equations :
(i) 4 sin² θ = 1
(ii) 3 tan² θ = 1.
Solution:

Question 10.
Find the principal value of x :

Question 11.
Find the principal value of x :
sec x = 2.
Solution:

Trigonometric Functions Long Answer Type Questions

Question 1.
Prove that:

Question 2.
Prove that:

Solution:

Question 3.
Prove that:

Question 4.
Prove that:

Question 5.
Prove that:

Solution:

Question 6.
Prove that:

Question 7.
Prove that:

Solution:

Question 8.
Prove that:

Question 9.
Prove that:

Solution:

Question 10.
Prove that:
tan 3A – tan 2A – tan A = tan 3A. tan 2A. tan A.
Solution:
tan3A – tan2A – tanA – tan3A.tan2A. tanA = 0
⇒ tan3A – tan2A – tanA (l + tan3A.tan2A) = 0
⇒ tan 3A – tan 2A = tan A (1 + tan 3 A. tan 2 A)

⇒ tan(3A – 2 A) = tan A
⇒ tanA = tan A.

Question 11.
Prove that:

Question 12.
Prove that:

Solution:

Question 13.
If α + β = π/4, then prove that :
(1 + tanα)(1 + tanβ) = 2.
Solution:
Given α + β = π/4
tan(α + β) = tan π/4

⇒ tan α + tan β = 1 – tanα.tan β
⇒ tan α + tan β.tanα.tan β = 1
⇒ tan α + tan β(1 + tanα) = 1
⇒ 1 + tan α + tan β(1 + tanα) = 1 + 1
⇒ ( 1 + tan α)(1 + tan β) = 2

Question 14.
Find the general solution of cos4x = cos 2x.
Solution:
Given equation :
cos4x = cos2x
⇒ cos4x – cos2x = 0

Question 15.

Question 16.
Solve the equation sin 3θ = sin 2θ
Solution:
Given equation sin3θ = sin2θ
sin 3θ – sin 2θ = 0

Question 18.
Solve the equation tanθ.tan2θ = 1
Solution:

Question 19.

Question 20.

Question 21.
Prove that:

Question 22.
Prove that:

Question 23.
prove that:
(cosx + cosy)²+ (sinx – siny)² = 4cos²x+y/2
Solution:

Question 24.
prove that:
(cosx – cosy)²+ (sinx – siny)² = 4sin²x–y/2
Solution:
L.H.S. = (cosx – cosy)²+ (sinx – siny)²
= cos²x + cos²y – 2 cosxcosy + sin²x + sin²y – 2sinx siny
= cos²x + sin²x + cos²y + sin²y – 2[cosx cosy + sinx siny]
= 1 + 1 – 2cos(x – y) = 2 – 2cos(x – y)
= 2[1 – cos(x – y)] = 2 x 2sin²
= 4 sin²x–y/2 = R.H.S.

Question 25.
Prove that:
sinx + sin3x + sin5x + sin7x = 4 cosx cos2x sin4x.
Solution:
L.H.S. = sinx + sin3x + sin5x + sin 7x
= sin7x + sinx + sin5x + sin3x

Question 26.
Prove that:

Solution:

Question 27.
Solve the equation and find general solution of sec²2x = 1 – tan2x. (NCERT)
Solution:
The given equation is :
sec² 2x = 1 – tan 2x
⇒ 1 + tan²2x = 1 – tan 2x
⇒ tan² 2x = – tan2x
⇒ tan² 2x + tan2x = 0 ⇒ tan2x (tan2x + 1) = 0
⇒ tan2x = 0 tan2x(tan2x + 1) = 0 ⇒ 2x = nπ or tan2x = – 1

Question 28.
Solve the equation 2 cos² x + 3 sin x = 0
Solution:
2 cos² x + 3 sin x = 0
⇒ 2 (1 – sin²  x)+3 sinx = 0
⇒ 2 – 2sin² x+ 3 sinx = 0
⇒ 2 sin² x – 3 sinx – 2 = 0
⇒ 2 sin² x + sinx – 4sinx – 2 = 0
⇒ sinx(2 sinx + 1) – 2(2sinx + 1) = 0
⇒ (2sinx + 1)(sinx – 2) = 0
⇒ 2 sin x +1 = 0 or sinx – 2 = 0
⇒ 2 sin x = – 1 = 0 or sinx = 2

Question 29.

Question 30.

Question 31.
Find the general solution of the equation sinx + sin5x + sin5x = 0
Solution:

Question 32.
Prove that:

Question 33.
Prove that:

Solution: