Trigonometry
- If sin α sec (30° + α) = 1 (0 < a < 60°), then the value of sin α + cos 2α is
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sinα = 1 cos(30° + α) ⇒ sinα = 1 sin(90° - 30 - α) ⇒ sinα = 1 sin(60° - α)
⇒ sin α = sin (60° – α)
⇒ 2α = 60° ⇒ α = 30°
∴ sinα + cos 2α
= sin 30° + cos 60°= 1 + 1 = 1 2 2
Correct Option: A
sinα = 1 cos(30° + α) ⇒ sinα = 1 sin(90° - 30 - α) ⇒ sinα = 1 sin(60° - α)
⇒ sin α = sin (60° – α)
⇒ 2α = 60° ⇒ α = 30°
∴ sinα + cos 2α
= sin 30° + cos 60°= 1 + 1 = 1 2 2
- If cos4 θ – sin4 θ = (2 / 3) , then the value of 2 cos²θ– 1 is
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cos4 θ - sin4 θ = 2 3 (cos2 θ + sin2 θ) (cos2 θ – sin2 θ) 2 3 ⇒ cos2 θ - sin2 θ 2 3 ⇒ 2cos2 θ - 1 2 3
Correct Option: C
cos4 θ - sin4 θ = 2 3 (cos2 θ + sin2 θ) (cos2 θ – sin2 θ) 2 3 ⇒ cos2 θ - sin2 θ 2 3 ⇒ 2cos2 θ - 1 2 3
- If sin α + cos β = 2 (0° ≤ β ≤ α ≤ 90°), then sin (2α + β / 3) =
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sinα + cosβ = 2
sinα ≤ 1 ; cosβ ≤ 1
⇒ α = 90° ; β = 0°= sin 60° = √3 2
Also,cos α = cos 30° = √3 3 2
Correct Option: B
sinα + cosβ = 2
sinα ≤ 1 ; cosβ ≤ 1
⇒ α = 90° ; β = 0°= sin 60° = √3 2
Also,cos α = cos 30° = √3 3 2
- If A, B and C be the angles of a triangle, then put of the following, the incorrect relation is :
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A + B + C = π
⇒ A + B = π - C 2 2 2 ⇒ sin A + B 2 = sin π - C 2 2 = cos C 2
Similarly,cos A + B = sin C 2 2 cot A + B = tan C 2 2 tan A + B = cot C 2 2
Correct Option: C
A + B + C = π
⇒ A + B = π - C 2 2 2 ⇒ sin A + B 2 = sin π - C 2 2 = cos C 2
Similarly,cos A + B = sin C 2 2 cot A + B = tan C 2 2 tan A + B = cot C 2 2
- If 0 < x < (π / 2) and secx = cosecy, then the value of sin (x + y) is :
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secx = cosecy
⇒ cosx = siny⇒ sin π - x = siny 2 ⇒ y = π - x 2 ⇒ x + y = π 2 ∴ sin (x + y) = sin π = 1 2
Correct Option: B
secx = cosecy
⇒ cosx = siny⇒ sin π - x = siny 2 ⇒ y = π - x 2 ⇒ x + y = π 2 ∴ sin (x + y) = sin π = 1 2