Question

Given that sin x =2/3 for x in quadrant 4, what are the values of a. sin2xb. cos2xc. tan2x

Answer 1

sin x = 2/3

opposite side = a = 2

hypotenuse = c= 3

adjacent side = b = ?

b^2 = 9 - 4

b = square root of 5

`\text{ b =- }\sqrt[]{5}`

Second quadrant

sin(2x) = 2sinxcosx

`\begin{gathered} \sin (2x)\text{ = 2(2/3)(}\frac{-\sqrt[]{5}}{3}) \\ \sin (2x)\text{ = -}(4)/(3)\cdot\text{ }\frac{\sqrt[]{5}}{3} \\ \sin (2x)\text{ = -}\frac{4\sqrt[]{5}}{9} \end{gathered}`

`\begin{gathered} \cos (2x)\text{ = }cos^2x-sin^2x \\ \cos (2x)\text{ = (}\frac{-\sqrt[]{5}}{3})^2\text{ - (}(2)/(3))^2 \\ \cos (2x)\text{ = }(5)/(9)\text{ - }(4)/(3) \\ \cos (2x)\text{ = }(5)/(9)\text{ - }(36)/(9) \\ \cos (2x)\text{ = }(-31)/(9) \end{gathered}`
`\text{ tan (2x) = 2}\frac{2}{\sqrt[]{5}}\text{ = }\frac{4}{\sqrt[]{5}}\text{ = }\frac{4\sqrt[]{5}}{5}`