trigonometry finding 2 possible $\cos$ values of an angle

Question

In a triangle $ABC$, $AB = 10$ cm and $AC= 5$ cm. The area is $15$ cm${}^2$ and the angle $BAC$ is equal to $\theta$.

Give two possible values of $\cos(\theta)$.

I was able to find one of the values of $\cos(\theta)$ which is 0.8 but how can I get a second value for $\cos(\theta)$?

Answer 1

The area of the triangle will be equal to $$\frac{(AB)(AC)\sin\theta}{2}$$ meaning that $\sin\theta=0.6$, as you probably found. This means that $$\cos^2\theta=1-\sin^2\theta=0.64$$ and $$\cos\theta=\color{green}{\pm}0.8$$ The positive value corresponds to $\theta$ an acute angle, and then negative value corresponds to obtuse $\theta$.