Trigonometry
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If α + θ = 7π and tanθ = √3 , then the value of tanθ is : 12
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tanθ = √3 = tan π 3 ⇒ θ = π 3 ∴ α + θ = 7π 12 ⇒ α = 7π - π 12 3 ⇒ α = 7π - π = π 12 4 ∴ tanα = tan π = 1 4
Correct Option: B
tanθ = √3 = tan π 3 ⇒ θ = π 3 ∴ α + θ = 7π 12 ⇒ α = 7π - π 12 3 ⇒ α = 7π - π = π 12 4 ∴ tanα = tan π = 1 4
- If cosθ + secθ = √3 , then the value of (cos3θ + sec3θ) is :
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a3 + b3 = (a + b)3 - 3ab(a + b)
∴ cos3θ + sec3θ = (cosθ + secθ)3 - 3cosθ . secθ(cosθ + secθ)
cos3θ + sec3θ = ( √3 )3 - 3 × √3
{ ∵ cosθ + secθ = √3 }
cos3θ + sec3θ = 3√3 - 3√3 = 0Correct Option: C
a3 + b3 = (a + b)3 - 3ab(a + b)
∴ cos3θ + sec3θ = (cosθ + secθ)3 - 3cosθ . secθ(cosθ + secθ)
cos3θ + sec3θ = ( √3 )3 - 3 × √3
{ ∵ cosθ + secθ = √3 }
cos3θ + sec3θ = 3√3 - 3√3 = 0
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The value of the following is : sinθ cosecθ tanθ cotθ sin2θ + cos2θ
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Expression = (sinθ cosecθ) (tanθ cotθ) sin2θ + cos2θ Expression = 1 × 1 = 1 1
{ ∴ sin2θ + cos2θ = 1 }Correct Option: A
Expression = (sinθ cosecθ) (tanθ cotθ) sin2θ + cos2θ Expression = 1 × 1 = 1 1
{ ∴ sin2θ + cos2θ = 1 }
- The value of (sec245° – cot245°) – (sin230° + sin260°) is
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sec245° – cot245° – sin230° – sin260° = (√2)2 - 1 - 
1 
2 - 1 2 2 2 sec245° – cot245° – sin230° – sin260° = 2 - 1 - 1 - 3 4 4 sec245° – cot245° – sin230° – sin260° = 1 - 1 - 3 4 4 sec245° – cot245° – sin230° – sin260° = 3 - 3 = 0 4 4
Correct Option: C
sec245° – cot245° – sin230° – sin260° = (√2)2 - 1 - 1 2 - 1 2 2 2 sec245° – cot245° – sin230° – sin260° = 2 - 1 - 1 - 3 4 4 sec245° – cot245° – sin230° – sin260° = 1 - 1 - 3 4 4 sec245° – cot245° – sin230° – sin260° = 3 - 3 = 0 4 4
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If sin 31° = x The value of sec 31° – sin 59° is y
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sin 31° = x y
∴ cos 31° = √1 - sin2 31°
cos 31° = √1 - (x / y)2⇒ cos 31° = √ (y2- x2) y2 ⇒ cos 31° = √(y² - x²) y ∴ sec 31° = y √(y² - x²)
∴ sec 31° - sin 59° = sec 31° - sin( 90 - 31° ) = sec 31° - cos 31°sec 31° - cos 31° = y - √(y² - x²) √(y² - x²) y sec 31° - cos 31° = y² - (y² - x²) y√(y² - x²) sec 31° - cos 31° = y² - y² + x² y√(y² - x²) sec 31° - cos 31° = x² y√(y² - x²)
Correct Option: A
sin 31° = x y
∴ cos 31° = √1 - sin2 31°
cos 31° = √1 - (x / y)2⇒ cos 31° = √ (y2- x2) y2 ⇒ cos 31° = √(y² - x²) y ∴ sec 31° = y √(y² - x²)
∴ sec 31° - sin 59° = sec 31° - sin( 90 - 31° ) = sec 31° - cos 31°sec 31° - cos 31° = y - √(y² - x²) √(y² - x²) y sec 31° - cos 31° = y² - (y² - x²) y√(y² - x²) sec 31° - cos 31° = y² - y² + x² y√(y² - x²) sec 31° - cos 31° = x² y√(y² - x²)
