Question

If cos x= √2/2 and x is fourth quadrant angle, evaluate sin 2x

Answer 1

First, we'll find sinx using the fundamental trigonometric relation that states that:

sin²x + cos²x = 1

`\begin{gathered} \sin ^2x+(\frac{\sqrt[]{2}}{2})^2=1 \\ \sin ^2x=1-(2)/(4) \\ \sin ^2x=(4)/(4)-(2)/(4) \\ \sin ^2x=(2)/(4) \\ \sin x=\pm\frac{\sqrt[]{2}}{2} \end{gathered}`

In the fourth quadrant, cosx is positive, and sinx is negative. So,

`\sin x=-\frac{\sqrt[]{2}}{2}`

We have another trigonometrical relation that states that

`\begin{gathered} \sin 2x=2\cdot\sin x\cdot\cos x \\ \sin 2x=2\cdot-\frac{\sqrt[]{2}}{2}\cdot\frac{\sqrt[]{2}}{2} \\ \sin 2x=(-2\cdot2)/(2\cdot2) \\ \sin 2x=-1 \end{gathered}`

Answer: Sin2x = -1