3 cot? A +2 sin? A
27. If sec A = 12, find :
tan? A - cos' A
28. If 5 cos 0 = 3, evaluate :
cosec 0 - cot e
cosec 0 + cot e
find the value of
29. If cosec A + sin A = 5
5:
cosec2 A + sin? A.
Answers
Answers:-
27. Given:
sec A = √2
√2 can be written as sec 45°.
→ sec A = sec 45°
On comparing both sides we get,
→ A = 45°
Hence,
→ (3 cot² A + 2 sin² A ) / (tan² A - cos² A)
= (3 cot² 45° + 2 sin² 45°) / (tan² 45° - cos² 45°)
- cot 45° = 1
- sin 45° = 1/√2
- tan 45° = 1
- Cos 45° = 1/√2
= [ 3 (1)² + 2(1/√2)²] / [ (1)² - (1/√2)² ]
= (3 * 1 + 2 * 1/2 ) / (1 - 1/2)
= [(3 + 1)] / [ (2 - 1) / 2]
= 4 * 2 / 1
= 8
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28. ("Theta" is taken as "A")
Given:
5 Cos A = 3
→ Cos A = 3/5
We know that,
Cos A = Adjacent side/ Hypotenuse
→ Adjacent side / Hypotenuse = 3/5
Using Pythagoras Theorem,
→ (Hypotenuse)² = (Opposite side)² + (Adjacent side)²
→ (5)² = (Opposite side)² + (3)²
→ (Opposite side)² = 25 - 9
→ (Opposite side)² = 16
→ Opposite side = √16
→ Opposite side = 4
Hence,
Cot A = Adjacent side/ Opposite side
→ cot A = 3/4
Cosec A = Hypotenuse/Opposite side
→ Cosec A = 5/3
Hence,
(Cosec A - cot A) / (Cosec A + cot A) = (3/4 - 5/3) / (3/4 + 5/3)
→ (Cosec A - cot A) / (Cosec A + cot A) = [(9 - 20) / 12 ] / [ (9 + 20) / 12 ]
→ (Cosec A - cot A) / (Cosec A + cot A) = ( - 11 / 12) * (12 / 29)
→ (Cosec A - cot A) / (Cosec A + cot A) = - 11 / 29
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29. Given:
Cosec A + sin A = 5
On squaring both sides we get,
→ (Cosec A + sin A)² = (5)²
Using the identity (a + b)² = a² + b² + 2ab in LHS we get,
→ cosec² A + sin² A + 2 * Cosec A * sin A = 25
Using Cosec A = 1/sin A in LHS we get,
→ cosec² A + sin² A + 2 * 1/sin A * sin A = 25
→ cosec² A + sin² A + 2(1) = 25
→ cosec² A + sin² A = 25 - 2