Answer to Question #123140 in Trigonometry for Ethel omelanga

Question

Express   Sin Cos y 8 6   in the form       rSin y , where  is an acute angle in
degrees.

Answer 1

According to the Trignometric identity

R Sin(θ+α) = a Sinθ + b Cosθ

We can write

r sin(y+α) = a siny + b cosy

we have a=8 , b=6

So,

r sin(y+α) = 8 siny + 6 cosy (1)

We can expand r sin(y+α)

r sin(y+ α) = r siny cosα + r sinα cosy (2)

now put equation (2) in equation (1)

r sin(y+ α) = 8 siny + 6 cosy

r siny cosα + r sinα cosy = 8 siny + 6 cosy

now comparing both sides

we got

r cosα =8 (3)

r sinα =6 (4)

now for acute angle

divide equation (4) by equation (3)

r sinα / r cosα = 6/8

tanα = 6/8

α=arc tan6/8

From equations (3), (4) it follows that $r^2\cos^2(\alpha)+r^2\sin^2(\alpha)=r^2=6^2+8^2=100,$ hence $r=8.$