Trigonometry
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If x cos²30°. sin 60° = tan² 45° . sec 60° then the value of x is cosec 60°
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x.cos²30° . sin 60°
= tan²45°.sec60° cosec 60° 
⇒ x × 3 × √3 = √3 4 2 ⇒ x = √3 × 8 = 8 = 2 2 3√3 3 3
Correct Option: C
x.cos²30° . sin 60°
= tan²45°.sec60° cosec 60° ⇒ x × 3 × √3 = √3 4 2 ⇒ x = √3 × 8 = 8 = 2 2 3√3 3 3
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If sin (θ + 30°) = 3 , then the value of cos²θ is √12
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sin (θ + 30°) = 3 √12 = 3 = √3 √2√3 2
⇒ sin (θ + 30°) = sin 60°
⇒ θ + 30° = 60°
⇒ θ = 60 – 30 = 30°
∴ cos²θ = cos²30°= 
√3 
² = 3 2 4
Correct Option: C
sin (θ + 30°) = 3 √12 = 3 = √3 √2√3 2
⇒ sin (θ + 30°) = sin 60°
⇒ θ + 30° = 60°
⇒ θ = 60 – 30 = 30°
∴ cos²θ = cos²30°= √3 ² = 3 2 4
- If 0 ≤ θ ≤ 90° and 4 cos²θ – 4 √3 cos θ + 3 = 0 then the value of θ is
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4 cos²θ – 4√3 cosθ + 3 = 0
⇒ (2cosθ)2 – 2.2 cosθ. √3 + (√3)² = 0
⇒ (2cosθ – √3)² = 0
⇒ 2 cosθ – √3 = 0
⇒ 2 cosθ = √3⇒ cosθ = √3 = cos 30° 2
⇒ θ = 30°Correct Option: A
4 cos²θ – 4√3 cosθ + 3 = 0
⇒ (2cosθ)2 – 2.2 cosθ. √3 + (√3)² = 0
⇒ (2cosθ – √3)² = 0
⇒ 2 cosθ – √3 = 0
⇒ 2 cosθ = √3⇒ cosθ = √3 = cos 30° 2
⇒ θ = 30°
- If sec θ – cos θ = (3 / 2) where θ is a positive acute angle, then the value of secθ is
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secθ – cosθ = 3 2 ⇒ secθ – 1 = 3 secθ 2 ⇒ sec²θ – 1 = 3 secθ 2
⇒ 2sec²θ – 2 = 3 secθ
⇒ 2 sec²θ – 3 secθ – 2 = 0
⇒ 2 sec²θ – 4 secθ + secθ – 2 = 0
⇒ 2 secθ (secθ – 2) + 1 (secθ – 2) = 0
⇒ (2 secθ + 1) (secθ – 2) = 0
⇒ secθ = 2 because 2 secθ + 1 ≠ 0
θ is positive acute angle.Correct Option: C
secθ – cosθ = 3 2 ⇒ secθ – 1 = 3 secθ 2 ⇒ sec²θ – 1 = 3 secθ 2
⇒ 2sec²θ – 2 = 3 secθ
⇒ 2 sec²θ – 3 secθ – 2 = 0
⇒ 2 sec²θ – 4 secθ + secθ – 2 = 0
⇒ 2 secθ (secθ – 2) + 1 (secθ – 2) = 0
⇒ (2 secθ + 1) (secθ – 2) = 0
⇒ secθ = 2 because 2 secθ + 1 ≠ 0
θ is positive acute angle.
- If tan (5x – 10°) = cot (5y + 20°), the value of (x + y) is
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tan (5x – 10°) = cot (5y + 20°)
⇒ tan (5x – 10°) = tan (90° – (5y + 20°))
⇒ 5x – 10° = 90° – 5y – 20°
⇒ 5x + 5y =70° + 10°
⇒ 5 (x + y) = 80°⇒ x + y = 80° = 16° 5
Correct Option: B
tan (5x – 10°) = cot (5y + 20°)
⇒ tan (5x – 10°) = tan (90° – (5y + 20°))
⇒ 5x – 10° = 90° – 5y – 20°
⇒ 5x + 5y =70° + 10°
⇒ 5 (x + y) = 80°⇒ x + y = 80° = 16° 5
