Question

If theta is in quadrant II and cos theta= -3/5, what is sin 2 theta + cos 2 theta?

Answer 1

ANSWER

sin(2θ) + cos(2θ) = -31/25 = -1.24

EXPLANATION

If we do the inverse of the cosine to -3/5 we would get the angle θ. Then we can know the value of the sine:

$\sin \theta=\sin (\cos ^{-1}(-\frac{3}{5}))=\frac{4}{5}$

So we have:

• sin(θ) = 4/5

,

• cos(θ) = -3/5

To find sin(2θ) + cos(2θ) we'll have to use the trigonometric identities:

$\begin{gathered} \sin 2\theta=2\sin \theta\cos \theta \\ \cos 2\theta=1-2\sin ^2\theta \end{gathered}$

Since we have the sine and cosine of theta, we can solve this:

$\sin 2\theta=2\cdot\frac{4}{5}\cdot(-\frac{3}{5})=-\frac{24}{25}$$\cos 2\theta=1-2(\frac{4}{5})^2=1-2\cdot\frac{16}{25}=1-\frac{32}{25}=-\frac{7}{25}$

The sum is:

$\sin 2\theta+\cos 2\theta=-\frac{24}{25}-\frac{7}{25}=-\frac{31}{25}=-1.24$