Question

If θ is a first quadrant angle in standard position with P(u,v) = (3,4) evaluate cos 2 θ

Answer 1

We have to evaluate cos(2θ), knowing that θ is in the first quadrant and in standard position P(u,v) = (3,4).

We can picture this as:

We can write the relation:

$\tan \theta=\frac{4}{3}$

We now look at the identities to find cos(2θ):

$\cos (2\theta)=\frac{1-\tan^2(2\theta)}{1+\tan^2(2\theta)}$

There are many identities for cos(2θ), but this is expressed in the information we already know, so we can solve as:

$\begin{gathered} \cos (2\theta)=\frac{1-\tan^2(2\theta)}{1+\tan^2(2\theta)} \\ \cos (2\theta)=\frac{1-(\frac{4}{3})^2}{1+(\frac{4}{3})^2} \\ \cos (2\theta)=\frac{1-\frac{16}{9}}{1+\frac{16}{9}} \\ \cos (2\theta)=\frac{\frac{9-16}{9}}{\frac{9+16}{9}} \\ \cos (2\theta)=\frac{-7}{25} \end{gathered}$

Answer: cos(2θ) = -7/25