Answer:
sin2x = 16/65
cos2x = -63/65
tan2x = -16/63
Explanation:
the tangent of an angle is equal to the opposite side over the adjacent side. So, if the tan(x) = 8, we can represent this as the following diagram:
Therefore, we can calculate the value of the hypotenuse as:
[tex]\text{Hypotenuse = }\sqrt[]{8^2+1^2}=\sqrt[]{64+1}=\sqrt[]{65}[/tex]With the hypotenuse, we can calculate sin(x) and cos(x) as follows:
[tex]\begin{gathered} \sin x=\frac{Opposite}{hypotenuse}=-\frac{8}{\sqrt[]{65}} \\ \cos x=\frac{\text{Adjacent}}{\text{hypotenuse}}=-\frac{1}{\sqrt[]{65}} \end{gathered}[/tex]We type the negative sign because the question says that sin(x) is negative.
Now, we will use the following trigonometric identities to find sin(2x), cos(2x) and tan(2x)
[tex]\begin{gathered} \sin 2x=2\sin x\cos x \\ \cos 2x=1-2\sin ^2x \\ \tan 2x=\frac{2\tan x}{1-\tan ^2x} \end{gathered}[/tex]Therefore, replacing the values, we get:
[tex]\sin 2x=2(\frac{-8}{\sqrt[]{65}})(\frac{-1}{\sqrt[]{65}})=\frac{16}{65}[/tex][tex]\cos 2x=1-2(\frac{-8}{\sqrt[]{65}})^2=1-2(\frac{64}{65})=-\frac{63}{65}[/tex][tex]\tan 2x=\frac{2(8)}{1-8^2}=-\frac{16}{63}[/tex]So, the answers are:
sin2x = 16/65
cos2x = -63/65
tan2x = -16/63
