Question

Sin 30x+sin 5x/cos 3x+cos 5x at x=π/16​

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Answer 1

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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Answer 2

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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