Trigonometrical Identities — Exercise 21(D)

If 2 sin θ = $\sqrt{3}$, then :

1. 2 cos θ + 1 = 0

2. 1 - sin θ = 0

3. cos θ = $\sqrt{3}$

4. 2 cos θ = 1

Answer

Given,

⇒ 2 sin θ = $\sqrt{3}$

⇒ sin θ = $\dfrac{\sqrt{3}}{2}$

⇒ sin θ = sin 60°

⇒ θ = 60°

⇒ cos θ = cos 60°

⇒ 2 cos θ = 2 cos 60°

⇒ 2 cos θ = $2 \times \dfrac{1}{2}$

⇒ 2 cos θ = 1.

Hence, Option 4 is the correct option.

If cos 62° 40' = x, the value of 1 + sin² 27° 20' is :

1. 1 + x²

2. 1 - x²

3. x² - 1

4. 1 + x

Answer

Given,

⇒ cos 62° 40' = x

⇒ cos (90° - 27° 20') = x

⇒ sin 27° 20' = x

Substituting value of sin 27° 20' in 1 + sin² 27° 20', we get :

⇒ 1 + sin² 27° 20' = 1 + x².

Hence, Option 1 is the correct option.

sin θ = 0.5 implies tan θ + cot θ is equal to :

1. 2 : $\sqrt{3}$

2. $\sqrt{3}$ : 4

3. 4 : $\sqrt{3}$

4. $\sqrt{3}$ : 2

Answer

Given,

⇒ sin θ = 0.5

⇒ sin θ = $\dfrac{5}{10}$

⇒ sin θ = $\dfrac{1}{2}$

⇒ sin θ = sin 30°

⇒ θ = 30°.

Substituting value of θ in tan θ + cot θ, we get :

⇒ tan θ + cot θ

⇒ tan 30° + cot 30°

⇒ $\dfrac{1}{\sqrt{3}} + \sqrt{3}$

⇒ $\dfrac{1 + 3}{\sqrt{3}}$

⇒ $\dfrac{4}{\sqrt{3}}$

⇒ 4 : $\sqrt{3}$.

Hence, Option 3 is the correct option.

If cos 63° 36' = 0.4446, the value of sin 26° 24' is :

1. 1

2. 1 + 0.4446

3. 0.4446

4. 1 - 0.4446

Answer

Given,

⇒ cos 63° 36' = 0.4446

⇒ cos (90° - 26° 24') = 0.4446

⇒ sin 26° 24' = 0.4446

Hence, Option 3 is the correct option.

Use tables to find sine of :

(i) 21°

(ii) 34° 42'

(iii) 47° 32'

(iv) 62° 57'

Answer

(i) From table,

sin 21° = 0.3584

Hence, sin 21° = 0.3584

(ii) From table,

sin 34° 42' = 0.5693

Hence, sin 34° 42' = 0.5693

(iii) From table,

sin 47° 30' = 0.7373

Difference of 2' = 0.0004       [To add]

∴ sin 47° 32' = 0.7373 + 0.0004 = 0.7377

Hence, sin 47° 32' = 0.7377

(iv) From table,

sin 62° 54' = 0.8902

Difference of 3' = 0.0004       [To add]

∴ sin 62° 57' = 0.8902 + 0.0004 = 0.8906

Hence, sin 62° 57' = 0.8906

Use tables to find cosine of :

(i) 2° 4'

(ii) 8° 12'

(iii) 26° 32'

(iv) 65° 41'

Answer

(i) From table,

cos 2° 6' = 0.9993

Difference of 2' = 0       [To add]

∴ cos 2° 4' = 0.9993 + 0 = 0.9993

Hence, 2° 4' = 0.9993

(ii) From table,

cos 8° 12' = 0.9898

Hence, 8° 12' = 0.9898

(iii) From table,

cos 26° 30' = 0.8949

Difference of 2' = 0.0003       [To subtract]

∴ cos 26° 32' = 0.8949 - 0.0003 = 0.8946

Hence, 26° 32' = 0.8946

(iv) From table,

cos 65° 42' = 0.4115

Difference of 1' = 0.0003       [To add]

∴ cos 65° 41' = 0.4115 + 0.0003 = 0.4118

Hence, 65° 41' = 0.4118

Use trigonometric tables to find tangent of :

(i) 37°

(ii) 42° 18'

Answer

(i) From table,

tan 37° = 0.7536

Hence, tan 37° = 0.7536

(ii) From table,

tan 42° 18' = 0.9099

Hence, tan 42° 18' = 0.9099

Use tables to find the acute angle θ, if the value of sin θ is :

(i) 0.4848

(ii) 0.3827

Answer

(i) Let sin θ = 0.4848

From table,

sin 29° = 0.4848

⇒ θ = 29°.

Hence, θ = 29°.

(ii) Let sin θ = 0.3827

From graph,

sin 22° 30' = 0.3827

θ = 22° 30'.

Hence, θ = 22° 30'.

Use tables to find the acute angle θ, if the value of cos θ is :

(i) 0.9848

(ii) 0.9574

Answer

(i) Let cos θ = 0.9848

From graph,

cos 10° = 0.9848

θ = 10°.

Hence, θ = 10°.

(ii) Let cos θ = 0.9574

From graph,

cos 16° 48' = 0.9573

cos θ - cos 16° 48' = 0.9574 - 0.9573 = 0.0004

From table,

Difference of 1' = 0.0004 [To subtract]

∴ θ = 16° 48' - 1' = 16° 47'.

Hence, θ = 16° 47'.

Use tables to find the acute angle θ, if the value of tan θ is :

(i) 0.2419

(ii) 0.4741

Answer

(i) Let tan θ = 0.2419

From table,

tan 13° 36' = 0.2419

θ = 13° 36'.

Hence, θ = 13° 36'.

(ii) Let tan θ = 0.4741

From table,

tan 25° 18' = 0.4727

tan θ - tan 25° 18' = 0.4741 - 0.4727 = 0.0014

From table,

Difference of 4' = 0.0014

∴ θ = 25° 18' + 4' = 25° 22'.

Hence, θ = 25° 22'.