Question

If sin(theta) = 3/5 and theta terminates in the first quadrant, find the exact value of cos (2 theta)

Answer 1

According to the problem, we have

`\sin \theta=(3)/(5)`

Then, we use the Pythagorean Trigonometric Identity

`\begin{gathered} (\sin \theta)^2+(\cos \theta)^2=1 \\ ((3)/(5))^2+(\cos \theta)^2=1 \\ (9)/(25)+(\cos \theta)^2=1 \\ (\cos \theta)^2=1-(9)/(25)=(25-9)/(25)=(16)/(25) \\ \cos \theta=(4)/(5) \end{gathered}`

Now, we use the Double Angle Identity

`\begin{gathered} \cos ^{}2\theta=\cos ^2\theta-\sin ^2\theta \\ \cos 2\theta=(16)/(25)-(9)/(25)=(16-9)/(25)=(7)/(25) \end{gathered}`

Hence, the exact value of cos (2 theta) is 7/25.