## Book: RD Sharma - Mathematics

### Chapter: 7. Values of Trigonometric Functions at Sum of Difference of Angles

#### Subject: - Class 11th

##### Q. No. 27 of Exercise 7.1

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27
##### If α,β are two different values of x lying between 0 And 2π which satisfy the equation 6 cos x + 8 sin x = 9, find the value of Sin(α+β).

Given 6 cos x + 8 sin x = 9

Case 1:

6 cos x = 9 – 8 sin x

Squaring on both sides,

36 cos2 x =(9 – 8 sin x)2

We know that cos2 x = 1 – sin2 x.

36(1 – sin2 x) = 81 + 64 sin2 x – 144 sin x

100 sin2 x – 144 sin x + 45 = 0

cos α And cos β are the roots of the a bove equation

sin α sin β = 45/100

Case 2:

8 sin x = 9 – 6 cos x

Squaring on both sides,

64 sin2 x =(9 – 6 cos x)2

We know that sin2 x = 1 – cos2 x

64(1 – cos2 x) = 81 + 36 cos2 x – 108 cos x

100 cos2 x – 108 cos x + 17 = 0

sin α And sin β are the roots of theAbove equation

cos α cos β = 17/100

Consider cos(α + β),

We know that cos(A +B) = cosA cosB - sinA sinB

We know that

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