Trigonometry
- If θ is a positive acute angle and cosec θ = √3 , then the value of cot θ – cosec θ is
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cosecθ = √3
cotθ = √cosec² θ - 1
=(√√3² - 1) = √3 - 1 = √2
∴ cotθ - cosecθ = √2 - √3= 3√2 - √2 = (√2 - √3) 3
Correct Option: E
cosecθ = √3
cotθ = √cosec² θ - 1
=(√√3² - 1) = √3 - 1 = √2
∴ cotθ - cosecθ = √2 - √3= 3√2 - √2 = (√2 - √3) 3
- If α and β are positive acute angles, sin (4α – β) = 1 and cos (2 α + β) = (1 / 2) , then the value of sin (α + 2β) is
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sin (4α – β) = 1 = sin 90°
⇒ 4α – β = 90° ...(i)cos (2α + β) = 1 = cos 60° 2
⇒ 2α + β = 60° ...(ii)
On adding equations (i) and (ii),
4α - β + 2α + β = 90° + 60°⇒ 6α = 150° ⇒ α = 150 = 25° 6
From equation (ii),
2 × 25 + β = 60°
⇒ β = 60° – 50° = 10°
∴ sin (α + 2β)
= sin (25 + 2 × 10)= sin 45° = 1 = 25° √2
Correct Option: D
sin (4α – β) = 1 = sin 90°
⇒ 4α – β = 90° ...(i)cos (2α + β) = 1 = cos 60° 2
⇒ 2α + β = 60° ...(ii)
On adding equations (i) and (ii),
4α - β + 2α + β = 90° + 60°⇒ 6α = 150° ⇒ α = 150 = 25° 6
From equation (ii),
2 × 25 + β = 60°
⇒ β = 60° – 50° = 10°
∴ sin (α + 2β)
= sin (25 + 2 × 10)= sin 45° = 1 = 25° √2
- If 0° < A < 90°, then the value of tan²A + cot² A – sec² A cosec² A is
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tan²A + cot²A – sec²A . cosec²A
= tan²A + cot²A – (1 + tan²A) (1 + cot²A)
= tan²A + cot²A – (1 + tan²A +
cot²A + cot²A.tan²A)
= tan²A + cot²A – 1 – tan²A – cot²A – cot²A . tan²A
= – 1 – 1 = – 2
[ tanA . cotA = 1]Correct Option: D
tan²A + cot²A – sec²A . cosec²A
= tan²A + cot²A – (1 + tan²A) (1 + cot²A)
= tan²A + cot²A – (1 + tan²A +
cot²A + cot²A.tan²A)
= tan²A + cot²A – 1 – tan²A – cot²A – cot²A . tan²A
= – 1 – 1 = – 2
[ tanA . cotA = 1]
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If cos θ = 3 then the value of sinθ . secθ . tanθ is 5
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cos θ = 3 5 ∴ sec θ = 5 3
∴ tan θ = √sec² θ - 1= 4 3 ∴ sinθ . secθ . tanθ = sin θ . tan θ cos θ = tan²θ = 4 ² = 16 3 9
Correct Option: B
cos θ = 3 5 ∴ sec θ = 5 3
∴ tan θ = √sec² θ - 1= 4 3 ∴ sinθ . secθ . tanθ = sin θ . tan θ cos θ = tan²θ = 4 ² = 16 3 9
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Expression
= 1 + sin θ + 1 - sin θ cos θ cos θ = 1 + sin θ + 1 - sin θ + 2 cos θ cos θ
= 2 secθCorrect Option: D
Expression
= 1 + sin θ + 1 - sin θ cos θ cos θ = 1 + sin θ + 1 - sin θ + 2 cos θ cos θ
= 2 secθ