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   √3/21/2  x ½

= 3/4 + 1/4

= 4/4

= 1

(ii) 2tan2 45° + cos2 30° − sin2 60°

= 2(1)2 + 3/43/4

= 2

(iii) (cos45°)/(sec30°+cosec30°)

(iv)

(v)

2. Choose the correct option and justify your choice.

(i) (2tan30˚)/­(1+tan230˚)

(a) sin 60˚

(b) cos 60˚

(d) sin30˚

(ii) (1-tan245˚)/(1+tan245˚)

(a) tan90˚

(b) 1˚

(d) 0˚

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

(a) 0˚

(b) 30˚

(d) 60˚

(iv) 2tan30˚/(1-tan230˚)

(a) cos60˚

(b) sin 60˚

(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.

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°

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.
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