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Introduction to Trigonometry

    Exercise 8.2

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10th class > Introduction to Trigonometry > Exercise 8.2

Exercise 8.2

(i)

Evaluate the following :

(i) sin 60° cos 30° + sin 30° cos 60°

= 32 32 + 12 12

= 34 + 14

= (3 + 1)4

= 44

=

(ii)

(ii) 2 tan2 45° + cos2 30° - sin2 60°

=212 + 322 - 322

= 2 + 34 - 34

=

(iii)

(iii) cos 45°(sec 30° + cosec 30°)

= 1223+2

= 122+233

= 1×32×2+23

= 3223+1

Multiplying numerator and denominator by 2 (3 - 1), we get

= 3223+1×23−123−1

= 32−643−1

= 32−68

(iv)

(iv) (sin 30° + tan 45° - cosec 60°)(sec 30° + cos 60° + cot 45°)

= 12+1−2323+12+1

= 32−2323+32

= 33−4234+3323

= 33−433+4

Multiplying numerator and denominator by 33 - 4, we get

= 33−433−433+433−4

= 27+16−24327−16

= 43−24311

(v)

(v) 5cos260°+4sec230°−tan245°sin230°+cos230°

=

= 54+163−114+34

= 15+64−12123+14

= 671244

=

(i)

Answer the following :

(i) 2 tan 30°1+tan230° = ?

Answer: sin °

(ii)

(ii) 1−tan245°1+tan245°= ?

Answer:

(iii)

(iii) sin 2A = 2 sin A is true when A =

Answer: °

(iv)

(iv) 2 tan 30°1−tan230°

Answer: tan °

3. If tan (A + B) = 3 and tan (A - B) = 13; 0° < (A + B) ≤ 90° ; A > B, find A and B

Solution:

Given that, tan (A + B) = 3 and, tan (A - B) = 13

Since, tan ° = 3 and tan ° = 13

Therefore, tan (A + B) = tan 60°

(A + B) = ° ...(i)

tan (A - B) = tan °

(A - B) = 30° ...(ii)

On adding both equations (i) and (ii), we obtain:

A + B + A - B = ° + 30°

A = °

A = °

By substituting the value of A in equation (i) we obtain

A + B = 60°

° + B = 60°

B = 60° - 45° = °

Therefore, ∠A = 45° and ∠B = 15° (A B).

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

(i)

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

Answer:

Let A = 30° and B = 60°

L.H.S = sin (A + B)

= sin (30° + °)

= sin °

=

R.H.S = sin A + sin B

= sin 30° + sin 60°

= 12 + 32

= 1+32

Since, sin (A + B) sin A + sin B.

Hence, the given statement is false.

(ii)

(ii) The value of sin θ increases from 0 to 1 as θ increases from 0° to 90°.

Answer:

sin 0° =

sin 30° = 12 =

sin 45° = 12 =

sin 60° = 32 =

sin 90° =

Hence, the given statement is true.

(iii)

(iii) The value of cos θ decreases from 1 to 0 as θ increases from 0° to 90°

Answer:

cos 0° =

cos 30° = 32 =

cos 45° = 12 =

cos 60° = 12 =

cos 90° =

Hence, the given statement is false.

(iv)

(iv) sin θ = cos θ for all values of θ, this is true when θ = 45°

Answer:

As sin ° = 12 and cos ° =12

It is not true for other values of θ

sin ° = 12 and cos ° = 32

sin ° = 32 and cos ° = 12

sin 90° = and cos 90° =

Hence, the given statement is false.

(v)

(v) cot A = cos Asin A

Answer:

cot 0° = cos 0°sin 0° = = undefined

Hence, the given statement is true.