Addition/subtraction formulas

[ruffnite]

Hi, I need help simplifying this expression:

(sin(x-30°) + cos(60°-x)) / (sinx)

What I got so far is expanding it to

(sinxcos30° - cosxsin30° + cos60°cosx + sin60°sinx) / (sinx)

I don't know how to proceed or if I was on the right track. Any help is appreciated thx!

 

[galactus]

Take note that sin(60)=cos(30) and so on. Use these as substitutions and it whittles down nicely.

Use the addition formulas for sin and cos.

 

[soroban]

Hello, ruffnite!

Hi, I need help simplifying this expression:

. . \(\displaystyle \frac{\sin(x-30) + \cos(60-x)}{\sin x}\)

What I got so far is expanding it to

. . \(\displaystyle \frac{\sin x\cos30 - \cos x\sin30 + \cos60\cos x + \sin60\sin x}{\sin x}\)

I don't know how to proceed or if I was on the right track. Any help is appreciated thx!

You're doing fine . . . just keeping going!

\(\displaystyle \text{We know that: }\;\begin{array}{cccccccc}\sin30^0 &=& \frac{1}{2} & & \sin60^o &=& \frac{\sqrt{3}}{2} \\ \\[-4mm]\cos30^o &=& \frac{\sqrt{3}}{2} & & \cos60^o &=& \frac{1}{2} \end{array}\)

\(\displaystyle \text{The fraction becomes: }\;\frac{\frac{\sqrt{3}}{2}\sin x - \frac{1}{2}\cos x + \frac{1}{2}\cos x + \frac{\sqrt{3}}{2}\sin x}{\sin x} \;=\;\frac{\sqrt{3}\sin x}{\sin x} \;=\;\sqrt{3}\)