Trigonometric functions

**Kluringen** · September 11th 2009, 02:26 PM

Hi!
In this problem I need to find all real numbers x.
$5sin(x)+cos(2x)=3$
and
$5sin(x)+cos(2x)=6$

Can you help me?

**Chris L T521** · September 11th 2009, 02:30 PM

Originally Posted by Kluringen

Hi!
In this problem I need to find all real numbers x.
$5sin(x)+cos(2x)=3$
and
$5sin(x)+cos(2x)=6$

Can you help me?

Recall that $\cos(2x)=\cos^2x-\sin^2x=1-2\sin^2x$ .

Thus, $5\sin x+\cos(2x)=3\implies 5\sin x+1-2\sin^2 x=3\implies2\sin^2 x-5\sin x+2=0$

Let $u=\sin x$ . Then solve $2u^2-5u+2=0$ , and then find $x$ .

Do something similar for the second one.

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