Question

Given that sin x =4/5 for x in quadrant 2, what are the values of a. sin2xb. cos2xc. tan2x

Answer 1

Simplify the equation to obain the value of x.

`\begin{gathered} \sin x=(4)/(5) \\ x=\sin ^(-1)((4)/(5)) \\ =53.130 \\ =180-53.130 \\ =126.87 \end{gathered}`

So value of x is 126.87, for x lies in second quadrant.

Determine the value of sin 2x.

`\begin{gathered} \sin (2\cdot126.87)=\sin (253.74) \\ =-0.96 \end{gathered}`

Determine the value of cos 2x.

`\begin{gathered} \cos (2\cdot126.87)=\cos 253.74 \\ =-0.2799 \\ \approx-0.28 \end{gathered}`

Determine the value of tan 2x.

`\begin{gathered} \tan (2\cdot126.87)=\tan (253.7) \\ =3.4286 \\ \approx3.43 \end{gathered}`

By using formula:

In second quadrant cos and tan have negative values.

Determine the value of sin 2x by using formula.

`\begin{gathered} \sin 2x=2\sin x\cos x \\ =2\sin x\cdot(-\sqrt[]{1-\sin ^2x}) \\ =-2\cdot(4)/(5)\cdot\sqrt[]{1-((4)/(5))^2} \\ =-(8)/(5)\cdot\sqrt[]{(25-16)/(25)} \\ =-(8)/(5)\cdot(3)/(5) \\ =-(24)/(25) \\ =-0.96 \end{gathered}`

Determine the value of cos 2x by using formula.

`\begin{gathered} \cos 2x=1-2\sin ^2x \\ =1-2\cdot((4)/(5))^2 \\ =1-(32)/(25) \\ =-(7)/(25) \\ =-0.28 \end{gathered}`

Determine the value of tan 2x by using formula.

`\begin{gathered} \tan 2x=(\sin 2x)/(\cos 2x) \\ =-(0.96)/(-0.28) \\ =3.4285 \\ \approx3.43 \end{gathered}`

So values of the expressions are,

sin 2x = -0.96

cos 2x = -0.28

tan 2x = 3.43