Trigonometry
- If cos x + cos y = 2, the value of sin x + sin y is
-
View Hint View Answer Discuss in Forum
cosx + cosy = 2
∵ cos x < 1
⇒ cos x = 1; cosy = 1
⇒ x = y = 0° [∵ Cos 0° = 1]
∴ sin x + sin y = 0Correct Option: A
cosx + cosy = 2
∵ cos x < 1
⇒ cos x = 1; cosy = 1
⇒ x = y = 0° [∵ Cos 0° = 1]
∴ sin x + sin y = 0
-
If tan θ + cot θ = 2, (0 ≤ θ ≤ 90°),then the value of sin θ is tan θ - cot θ
-
View Hint View Answer Discuss in Forum
tan θ + cot θ = 2 tan θ - cot θ 1
By componendo and dividendo,2 tan θ = 3 2 cot θ 1 ⇒ sin θ . sin θ = 3 cos θ cos θ
&rArr sin² θ = 3cos² θ
&rArr sin² θ = 3 (1 – sin² θ)
&rArr 4 sin² θ = 3⇒ sin² θ = 3 4 ⇒ sin θ = √3 2
Correct Option: B
tan θ + cot θ = 2 tan θ - cot θ 1
By componendo and dividendo,2 tan θ = 3 2 cot θ 1 ⇒ sin θ . sin θ = 3 cos θ cos θ
&rArr sin² θ = 3cos² θ
&rArr sin² θ = 3 (1 – sin² θ)
&rArr 4 sin² θ = 3⇒ sin² θ = 3 4 ⇒ sin θ = √3 2
- The value of tan 4°.tan 43°.tan 47°.tan 86° is
-
View Hint View Answer Discuss in Forum
tan 4°. tan 43°. tan 47°. tan 86°
= tan 4°. tan 43°. tan (90° – 43°).
tan (90° – 4°)
= tan 4° × tan 43° × cot 43° × cot 4° = 1
[ tan (90° – q) = cot q; tan q. cot q = 1]Correct Option: C
tan 4°. tan 43°. tan 47°. tan 86°
= tan 4°. tan 43°. tan (90° – 43°).
tan (90° – 4°)
= tan 4° × tan 43° × cot 43° × cot 4° = 1
[ tan (90° – q) = cot q; tan q. cot q = 1]
- If cosec 39° = x, the value of (1 / cosec² 51°) + sin² 39°+ tan² 51° – (1 / sin² 51° sec² 39° is
-
View Hint View Answer Discuss in Forum
1 + sin²39° + tan²51° - 1 cosec²51° sin²51°.sec²39° = sin²51° + sin²39° + tan²(90° – 39°) – 1 sin²(90° - 39°).sec²39°
[∵ sin (90° – θ) = cos θ
tan (90° – θ) = cot θ]
= 1 + cot²39° – 1
= cosec² 39° – 1 = x² - 1Correct Option: C
1 + sin²39° + tan²51° - 1 cosec²51° sin²51°.sec²39° = sin²51° + sin²39° + tan²(90° – 39°) – 1 sin²(90° - 39°).sec²39°
[∵ sin (90° – θ) = cos θ
tan (90° – θ) = cot θ]
= 1 + cot²39° – 1
= cosec² 39° – 1 = x² - 1
- The minimum value of 2 sin²θ+ 3 cos²θis
-
View Hint View Answer Discuss in Forum
2 sin²θ + 3cos²θ
= 2 sin²θ + 2cos²θ + cos²θ
= 2 (sin²θ + cos²θ) + cos²θ
= 2 + cos² θ [∵ sin² θ + cos²θ =1]
∵ Minimum value of cos θ = –1
But cos² θ > 0, when θ = 90°
[∵ cos 0° = 1, cos 90° = 0]
∴ Required minimum value = 2 + 0 = 2Correct Option: C
2 sin²θ + 3cos²θ
= 2 sin²θ + 2cos²θ + cos²θ
= 2 (sin²θ + cos²θ) + cos²θ
= 2 + cos² θ [∵ sin² θ + cos²θ =1]
∵ Minimum value of cos θ = –1
But cos² θ > 0, when θ = 90°
[∵ cos 0° = 1, cos 90° = 0]
∴ Required minimum value = 2 + 0 = 2
