Trigonometry
- What is the value of cos 105°?
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cos 105° = cos(60° + 45°)
= cos 60° × cos 45° – sin 60° × sin 45°
[∵ cos (A + B) = cosA cosB – sinA sinB]= 1 . 1 - √3 . 1 = 1 - √3 2 √2 2 √2 2√2 Correct Option: A
cos 105° = cos(60° + 45°)
= cos 60° × cos 45° – sin 60° × sin 45°
[∵ cos (A + B) = cosA cosB – sinA sinB]= 1 . 1 - √3 . 1 = 1 - √3 2 √2 2 √2 2√2
- The radian measure of 120° will be
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We know that,
1° = 
π 
R 180 ⇒ 120° = π × 120 R = 2π R 180 3 Correct Option: D
We know that,
1° = π R 180 ⇒ 120° = π × 120 R = 2π R 180 3
- If cosA + cos²A = 1, then the value of (sin²A + sin4A) is :
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cosA + cos²A = 1
⇒ cosA = 1 – cos²A = sin²A
∴ sin²A + sin4A
= sin²A + (sin²A)²
= sin²A + cos²A = 1Correct Option: D
cosA + cos²A = 1
⇒ cosA = 1 – cos²A = sin²A
∴ sin²A + sin4A
= sin²A + (sin²A)²
= sin²A + cos²A = 1
- If sinq+ cosecθ = 2, then the value of (sin-7θ + cosec7θ) is
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sinθ + cosecθ = 2
⇒ sinθ + 1/sinθ = 2
⇒ sin2θ + 1 = 2sinθ
⇒ sin²θ – 2sinθ + 1 = 0
⇒ (sinθ – 1)² = 0
⇒ sinθ – 1 = 0
⇒ sinθ = 1
∴ cosecθ = 1
∴ sin-7θ + cosec7θ
= (1)-7 + (1)7 = 2Correct Option: C
sinθ + cosecθ = 2
⇒ sinθ + 1/sinθ = 2
⇒ sin2θ + 1 = 2sinθ
⇒ sin²θ – 2sinθ + 1 = 0
⇒ (sinθ – 1)² = 0
⇒ sinθ – 1 = 0
⇒ sinθ = 1
∴ cosecθ = 1
∴ sin-7θ + cosec7θ
= (1)-7 + (1)7 = 2
- If 2y cosq = x sinθ and 2x secθ – y cosecθ = 3 then what is the value of (x² + 4y²) ?
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2y cosθ = x sinθ
⇒ x = 2y cosθ .....(i) sinθ
∴ 2x secθ – y cosecθ = 3⇒ 2 × 2 × ycosθ.secθ - ycosecθ = 3 sinθ
⇒ 4y cosecθ – y cosecθ = 3
⇒ 3y cosecθ = 3⇒ y = 1 = sinθ cosecθ
From equation (i),x = 2 × sinθ.cosθ = 2cosθ sinθ
&therae4; x² + 4y² = (2cosθ)² + 4 sin²θ
= 4 (cos²θ + sin²θ) = 4Correct Option: A
2y cosθ = x sinθ
⇒ x = 2y cosθ .....(i) sinθ
∴ 2x secθ – y cosecθ = 3⇒ 2 × 2 × ycosθ.secθ - ycosecθ = 3 sinθ
⇒ 4y cosecθ – y cosecθ = 3
⇒ 3y cosecθ = 3⇒ y = 1 = sinθ cosecθ
From equation (i),x = 2 × sinθ.cosθ = 2cosθ sinθ
&therae4; x² + 4y² = (2cosθ)² + 4 sin²θ
= 4 (cos²θ + sin²θ) = 4
