Thread: Proving Trig Identity: Sine Difference

1. Proving Trig Identity: Sine Difference

In this figure, the triangle has hypotenuse 1.

I need to use it to show that:

$
sin(A - B) = sinAcosB - sinBcosA
$

The hint is :
$
sin(A - B) = \frac{X}{h}
$

Find other expressions for X and h and substitute them into this equation.

I know this is easy. But I'm just stuck. I've tried working out other side lengths and looking at other trig identities but I'm just at a brick wall without a clue why.

Thanks for any help.

2. Answer

Greetings. You don't need any other trig identities. Everything is on the diagram

Step 1. Fill in the angles, and you'll see that the 3rd from left triangle and the large triangle are similar.

Step 2. Find the scale factor.
In this case, the large triangle has a hypotenuse of 1 and the small triangle has a hypotenuse of $sinA-hsinB$
Hence scale factor(Q): $Q=\frac{1}{sinA-hsinB}$

Step 3. Apply this to the other side of the triangle, opposite the angle of $(90-A)$.
$QX=cosA$
Hence: $X=cosAsinA-hcosAsinB$

Put into
And you get: $sin(A-B)=\frac{cosAsinA-hcosAsinB}{h}$

Simplify, then look at your triangle and realize that
$cosB=\frac{cosA}{h}$

Substitute and you arrive at the formula

3. Excellent solution I-think!

Cleared that right up.

Thank you so much.