Introduction to Trigonometry-Exercise 8.2-Class 10

1. Evaluate the following

(i) sin60° cos30° + sin30° cos 60°

(ii) 2tan² 45° + cos² 30° − sin² 60°

(iii)  ^(cos45°⁾/_(sec30°_+cosec30°₎

(iv)

(v)

Solution:

(i) sin60° cos30° + sin30° cos 60°

=  ^√3/₂   x  ^√3/₂ +  ¹/₂  x ^½

= ³/₄ + ¹/₄

= ⁴/₄

= 1

 

(ii) 2tan² 45° + cos² 30° − sin² 60°

= 2(1)² + ³/₄ – ³/₄

= 2

(iii) ^(cos45°⁾/_(sec30°_+cosec30°₎

 

(iv)

 

(v)

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2. Choose the correct option and justify your choice. 

(i) ^(2tan30˚)/_{­(1+tan230˚)}

(a) sin 60˚

(b) cos 60˚

© tan60˚

(d) sin30˚

 

(ii) (1-tan²45˚)/(1+tan²45˚)

(a) tan90˚

(b) 1˚

© sin45˚

(d) 0˚

 

(iii) sin2A = 2sinA is true when A =

(a) 0˚

(b) 30˚

© 45˚

(d) 60˚

 

(iv) 2tan30˚/(1-tan²30˚)

(a) cos60˚

(b) sin 60˚

©tan60˚

(d) sin30˚

Solution:

(i) ^(2tan30˚)/_{­(1+tan²30˚)}

Out of the given alternatives, only sin60˚= ^√3/₂ .

Hence, (A) is correct.

 

(ii) (1-tan²45˚)/(1+tan²45˚)

= [(1-(1)²]/ [(1+(1)²]

= (1-1)/(1+1)

= 0/2

= 0

Hence, (D) is correct.

 

(iii)sin2A = 2sinA is true when A =

Out of the given alternatives, only A = 0° is correct.

As sin 2A = sin 0° = 0

2sinA = 2sin 0° = 2(0) = 0. Hence, (A) is correct.

 

(iv) 2tan30˚/(1-tan²30˚)

Out of the given alternatives, only tan 60° = √3. Hence, (C) is correct.

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3. If tan(A+B) = √3 and tan(A-B) = ¹/_√3

0˚< A+B ≤ 90˚, A>B, find A and B.

Solution:

tan(A+B) = √3

tan(A+B) = tan60

A+B = 60 ———(1)

tan(A-B) = ¹/_√3

⇒tan (A − B) = tan30

⇒A − B = 30 ————–(2)

On adding both equations, we obtain 2A = 90

⇒ A = 45

From equation (1), we obtain

45 + B = 60

B = 15

Therefore, ∠A = 45° and ∠B = 15°

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4. State whether the following are true or false. Justify your answer.

(i) sin (A + B) = sin A + sin B

(ii) The value of sinθ increases as θ increases

(iii) The value of cos θ increases as θ increases

(iv) sinθ = cos θ for all values of θ

(v) cot A is not defined for A = 0° 

Solution:

(i) sin (A + B) = sin A + sin B

Let A = 30° and B = 60° ;

sin (A + B) = sin (30° + 60°) = sin 90° = 1

We have, sin A + sin B = sin 30° + sin 60°

¹/₂ + ^√3/₂ = ^(1+√3)/₂

Clearly, sin (A + B) ≠ sin A + sin B

Hence, the given statement is false.

 

(ii) The value of sin θ increases as θ increases in the interval of 0° < θ < 90° as sin 0° = 0

sin 30˚= ¹/₂ = 0.5

sin 45˚= ¹/_√2 = 0.707

cos60˚= ¹/₂ = 0.5

cos90˚ = 0

It can be observed that the value of cos θ does not increase in the interval of 0°<θ

<90°. Hence, the given statement is false. >

 

(iv) sin θ = cos θ for all values of θ.

This is true when θ = 45°

As sin˚45 = ¹/_√2

cos˚45 = ¹/_√2

It is not true for all other values of θ.

As sin30˚ = ¹/₂ and cos30 ˚ = ^√3/₂

Hence, the given statement is false.

 

(v) cot A is not defined for A = 0°

As CotA= cosA/sinA

cot0˚ = cos0˚/sin0˚ = ¹/₀ undefined

Hence, the given statement is true.