Trigonometry elementary problem

1. Jul 9, 2012

sambarbarian

1. The problem statement, all variables and given/known data
If 3 sin theta + 5cos theta = 5 ... prove that 5sin theta - 3 cos theta = +- 3

2. Relevant equations

3. The attempt at a solution

i tried many things here , mostly those including squares , because i need +-3 , but this one has me stumped :/

2. Jul 9, 2012

Curious3141

Are you familiar with this "trick": $a\sin \theta + b\cos \theta = \sqrt{a^2 + b^2}\sin(\theta + \arctan{\frac{b}{a}})$?

3. Jul 9, 2012

sambarbarian

never heard of it

4. Jul 9, 2012

Curious3141

It's basically the angle sum formula for sine. Try expanding $R\sin(\theta + \alpha)$ and compare coefficients to $a \sin \theta + b \cos \theta$.

5. Jul 9, 2012

sambarbarian

can this question be solved without it ?>

6. Jul 9, 2012

Curious3141

Not easily, I think. But why don't you want to try this? Haven't you covered the angle sum formula at all?

7. Jul 9, 2012

Saitama

You can try writing $5cosθ=5\sqrt{1-sin^2θ}$. Taking this term to RHS,
you will get an equation
$$3sinθ-5=-5\sqrt{1-sin^2θ}$$
Square both the sides, the equation will be easy to solve and you will get two values for θ.

8. Jul 9, 2012

SammyS

Staff Emeritus
Very good !

9. Jul 9, 2012

sambarbarian

thank you!!!! that did the trick

10. Jul 9, 2012

Saitama

Thank you SammyS!