Trigonometry
- If cos 21° = x/y , then (cosec 21° – cos 69°) is equal to
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cos 21° = x/y
∴ cos 69° = cos (90° – 21°)
= sin 21°= √1 - cos² 21° = √1 - x² = √y² - x² y² y ∴ cosec 21° = y √y² - x²
∴ cosec 21° – cos 69°= y = √y² - x² √y² - x² y = y² - (y² - x²) = x² y√y² - x² y√y² - x² Correct Option: A
cos 21° = x/y
∴ cos 69° = cos (90° – 21°)
= sin 21°= √1 - cos² 21° = √1 - x² = √y² - x² y² y ∴ cosec 21° = y √y² - x²
∴ cosec 21° – cos 69°= y = √y² - x² √y² - x² y = y² - (y² - x²) = x² y√y² - x² y√y² - x²
- If α + β = 90° and α : β = 2 : 1, then the ratio of cosα to cosβ is :
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α : β = 2 : 1
Sum of the terms of ratio
= 2 + 1 = 3
α + β = 90°∴ α = 2 × 90° = 60° 3
β = 30°∴ cosα = cos 60° = 1/2 cosβ cos 60° √3/2
= 1 : √3Correct Option: A
α : β = 2 : 1
Sum of the terms of ratio
= 2 + 1 = 3
α + β = 90°∴ α = 2 × 90° = 60° 3
β = 30°∴ cosα = cos 60° = 1/2 cosβ cos 60° √3/2
= 1 : √3
- If θ is positive acute angle and 7 cos²θ + 3 sin²θ = 4, then the value of θ is :
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7 cos²θ + 3 sin²θ = 4
⇒ 7 (1 – sin²θ) + 3 sin²θ = 4
⇒ 7 – 7 sin²θ + 3sin²θ = 4
⇒ 7 – 4 sin²θ = 4
⇒ 4 sin²θ = 7 – 4 = 3⇒ sin²θ = 3 4 ⇒ sinθ = √3 2
∵0 < q < 90°
⇒ θ = 60°Correct Option: A
7 cos²θ + 3 sin²θ = 4
⇒ 7 (1 – sin²θ) + 3 sin²θ = 4
⇒ 7 – 7 sin²θ + 3sin²θ = 4
⇒ 7 – 4 sin²θ = 4
⇒ 4 sin²θ = 7 – 4 = 3⇒ sin²θ = 3 4 ⇒ sinθ = √3 2
∵0 < q < 90°
⇒ θ = 60°
- If tanθ = 4/3, then the value of
3sinθ + 2cosθ is : 3sinθ - 2cosθ
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tanθ = 4 3 Expression = 3sinθ + 2cosθ 3sinθ - 2cosθ
On dividing numerator and denominator by cosθ,= 3sinθ + 2cosθ cosθ cos;θ 3sinθ - 2cosθ cosθ cosθ = 3tanθ + 2 3tanθ - 2 = 3 × 4 + 2 3 3 × 4 - 2 3 = 4 + 2 = 6 4 - 2 2 Correct Option: C
tanθ = 4 3 Expression = 3sinθ + 2cosθ 3sinθ - 2cosθ
On dividing numerator and denominator by cosθ,= 3sinθ + 2cosθ cosθ cos;θ 3sinθ - 2cosθ cosθ cosθ = 3tanθ + 2 3tanθ - 2 = 3 × 4 + 2 3 3 × 4 - 2 3 = 4 + 2 = 6 4 - 2 2
- If sec (4x – 50°) = cosec (50° – x), then the value of x is
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sec (4x – 50°) = sec (50° – x)
⇒ sec (4x – 50°) = sec (90° – (50° – x)) = sec (40° + x)
⇒ 4x – 50° = 40° + x
⇒ 4x – x = 50° + 40°
⇒ 3x = 90°⇒ x = 90° = 30° 3 Correct Option: C
sec (4x – 50°) = sec (50° – x)
⇒ sec (4x – 50°) = sec (90° – (50° – x)) = sec (40° + x)
⇒ 4x – 50° = 40° + x
⇒ 4x – x = 50° + 40°
⇒ 3x = 90°⇒ x = 90° = 30° 3
