Question

Find the exact value of each trigonometric function for the given angle θ.

Answer 1

`\sin (240^\circ)=-(√(3))/(2),\cos (240^\circ)=-(1)/(2),\tan (240^\circ)=√(3),\cot (240^\circ)=(1)/(√(3)),\sec (240^\circ)=-2,\csc (240^\circ)=(2)/(√(3)).`

Explanation:

The given angle is 240 degrees.

We need to find the exact value of each trigonometric function for the given angle θ.

Since
`\theta=240`, it means θ lies in 3rd quadrant. In 3d quadrant only tan and cot are positive.

`\sin (240^\circ)=\sin (180^\circ+60^\circ)=-\sin (60^\circ)=-(√(3))/(2)`

`\cos (240^\circ)=\cos (180^\circ+60^\circ)=-\cos (60^\circ)=-(1)/(2)`

`\tan (240^\circ)=\tan (180^\circ+60^\circ)=\tan (60^\circ)=√(3)`

`\cot (240^\circ)=\cot (180^\circ+60^\circ)=\cot (60^\circ)=(1)/(√(3))`

`\sec (240^\circ)=\sec (180^\circ+60^\circ)=-\sec (60^\circ)=-2`

`\csc (240^\circ)=\csc (180^\circ+60^\circ)=-\csc (60^\circ)=-(2)/(√(3))`

Therefore,
`\sin (240^\circ)=-(√(3))/(2),\cos (240^\circ)=-(1)/(2),\tan (240^\circ)=√(3),\cot (240^\circ)=(1)/(√(3)),\sec (240^\circ)=-2,\csc (240^\circ)=-(2)/(√(3)).`