If $\sin^3(\theta)+\cos^3(\theta) = \frac{11}{16}$, find the exact value of $\sin(\theta) + \cos(\theta)$

Question

The equation is $$\sin^3(\theta)+\cos^3(\theta) = \frac{11}{16}$$ and it wants me to find the exact value of $\sin(\theta) + \cos(\theta)$.

I started at first trying to use Pythagorean identities, but those only work for squared trigs. I also tried to expand/use foil, but I'm stuck; not sure if this method is even the right one to use.

Answer 1

Hint: let $s = \sin \theta, c = \cos \theta$.

What you want to find is $s+c$.

You're given $s^3 + c^3$

Now $s^3 + c^3 = (s+c)(s^2 + c^2 - sc)$

And $2sc = (s+c)^2 - (s^2 + c^2)$

Can you finish?