# Given isosceles triangle, find sin (A-C)

Tags:
1. Aug 24, 2015

### terryds

1. The problem statement, all variables and given/known data

Triangle ABC have side AB = 10 cm, AC=BC = 13cm, so sin (A-C) is...

2. Relevant Equations

sin (A-C) = sin A cos C - cos A sin C

3. The attempt at a solution

I see that the triangle can be split into two right-angle triangles.
But, sin (A-C) ?? How to get that?

2. Aug 24, 2015

### SteamKing

Staff Emeritus
Is there a sketch of this triangle?

3. Aug 24, 2015

### terryds

Actually, there is no sketch of the triangle in the question.
But, here is my sketch

4. Aug 24, 2015

### William White

use sohcahtoa

you know all the lengths, you can easily now work out all the angles.

http://www.mathwords.com/s/s_assets/s126.gif

5. Aug 24, 2015

### NickAtNight

Where did that equation come from?

6. Aug 24, 2015

### SammyS

Staff Emeritus
It's one of the well known angle sum/difference formulas.

7. Aug 24, 2015

### NickAtNight

True, but does he need that here?

8. Aug 24, 2015

### SammyS

Staff Emeritus
Is it necessary? I don't know.

However, it can be used to solve the problem

9. Aug 24, 2015

### NickAtNight

Seems like the wrong equation to apply to me.

Edit: Never mind. Now I see. The next step makes it a simple multiplication, division and subtraction problem. That is a very elegant solution when applied that way. Cause we can apply the lengths directly !

We use the equations WW posted, but rather than calculate the angles, we eliminate the Sin's and Cos's. to get a simple math expression! No trig tables or Scientific calculator needed.

Last edited: Aug 24, 2015
10. Aug 25, 2015

### terryds

So, should I use the formula sin (A-C) = sin A cos C - cos A sin C ??

sin A = 12/13
cos A = 5/13

sin (C/2) = 5/13
cos (C/2) = 12/13

Then,
sin (C) = 2 * sin (C/2) * cos (C/2) = 2 * (5/13) * (12/13) = 120/169
cos (C) = 1 - 2 sin^2(C/2) = 1 - 2 * 25/169 = 1 - 50/169 = 119/169

So,

sin (A - C) = sin A cos C - cos A sin C = 12/13 * 119/169 - 5/13*120/169 = 1428/2197 - 600/2197 = 828/2197

Do I get it right ?? Is there any simpler solution ?

11. Aug 25, 2015

### NickAtNight

Besides using a calculator with trig functions?
A = asin 12/13.
B=A
C = 180 - A - B

Dif = A-C

edited. And edited
Sin (A) = (12/13) = 0.923. A = Asin (1.17) = 67.38 degrees. Same as B. C= 180-2A = 45.23
A-C = 67.38 - 45.23 = 22.14

Last edited: Aug 25, 2015
12. Aug 25, 2015

### NickAtNight

By golly, it appears that you got it right ! (Took me a while to get the calculation right the other way)

Last edited: Aug 25, 2015
13. Aug 25, 2015

### terryds

Yap, using calculator is the simplest way, but it's not allowed in the test :(