Question

Given cos x =-8/17 and tan x <0 find sin2x

Answer 1

The value of cos x and tan x is less tan x is less than 0, means angle x lies in the second quadrant as in second quadrant cosine and tangent of an angle is negative.

Determine the measure of angle x from the equation cos x = -8/17.

`\begin{gathered} \cos x=-(8)/(17) \\ x=\cos ^(-1)(-(8)/(17)) \\ =118.07 \end{gathered}`

Substitute 118.07 for x in the expression sin 2x to obtain the value of sin 2x.

`\begin{gathered} \sin (2\cdot118.07)=\sin (236.14) \\ =-0.8304 \\ \approx-0.83 \end{gathered}`

Thus, value of sin 2x is -0.83.

OR

Determine the value of sin x by using trigonometry identity.

`\begin{gathered} \sin x=\sqrt[]{1-(\cos x)^2} \\ =\sqrt[]{1-(-(8)/(17))^2} \\ =\sqrt[]{(289-64)/(289)} \\ =(15)/(17) \end{gathered}`

Determine the value of sin 2x by using trigonometry identity.

`\begin{gathered} \sin 2x=2\sin x\cos x \\ =2\cdot(15)/(17)\cdot(-(8)/(17)) \\ =-(240)/(289) \end{gathered}`

So answer is -240/289.