# Task 11.1

For what values of $c$ are the numbers $\sin \alpha$ and $\cos \alpha$ the roots of the quadratic equation: $10 x^{2}-7 x-c=0$ (where $\alpha$ is some angle)?

## Number of points 7
Answer:

when $c=2.55$

## Solution

By Vieta's theorem $\sin \alpha+\cos \alpha=0.7$.

Then $1+2 \sin \alpha \cos \alpha=(\sin \alpha+\cos \alpha)^{2}=0.49$.

Therefore, $\sin \alpha \cos \alpha=-0.255$.

Thus, $c=-10 \sin \alpha \cos \alpha=2.55$.

The roots of the obtained equation are indeed the sine and cosine of some angle, since the equation $\sin \alpha+\cos \alpha=0.7$ clearly has roots.