sin 30x+sin 5x/cos 3x+cos 5x at x=π/16
Answers
Question :- find the value of sin 3x+sin 5x/cos 3x+cos 5x at x = π/16 ?
Solution :-
→ (sin 3x+sin 5x) /(cos 3x+cos 5x)
→ (sin 5x + sin 3x) /(cos 5x+cos 3x)
using :-
- sin C + sin D = 2 * sin(C + D/2) * cos(C - D)/2 in numerator .
- cos C + cos D = 2 * cos(C + D/2) * cos(C - D)/2 in denominator .
→ [2 * sin(5x + 3x/2) * cos(5x - 3x/2)] / [2 * cos(5x + 3x/2) * cos(5x - 3x/2)]
→ [2 * sin(8x/2) * cos(2x/2)] / [2 * cos(8x/2) * cos(2x/2)]
→ (2 * sin 4x * cos x) / (2 * cos 4x * cos x)
→ sin 4x / cos 4x
using :-
- tan A = sin A / cos A
→ tan 4x
putting value of x now,
→ tan 4(π/16)
→ tan (π/4)
→ tan 45°
→ 1 (Ans.)
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Given : (Sin3x + Sin5x)/(cos3x + cos5x)
To Find : Value at x=π/16
Solution:
(Sin3x + Sin5x)/(cos3x + cos5x)
Sin A + Sin B = 2 Sin(A + B)/2 Cos(A - B)/2
=> Sin3x + Sin5x = 2 Sin 4x Cos(-x)
Cos A + Cos B = 2 Cos(A + B)/2 Cos(A - B)/2
=>Cos3x +Cos5x = 2 Cos 4x Cos(-x)
(Sin3x + Sin5x)/(cos3x + cos5x)
= 2 Sin 4x Cos(-x) / 2 Cos 4x Cos(-x)
= tan 4x
x = π/16
= tan 4π/16
= tan π/4
= 1
(Sin3x + Sin5x)/(cos3x + cos5x) = 1 at x=π/16
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