Question

find the following trigonometric values. express your answers exactly. cos(150°) = sin(150°) =

Explanation:

Step1: Analyze the angle 150°

150° is in the second quadrant, and we can express it as \(180^{\circ}- 30^{\circ}\). We will use the reference angle concept and the trigonometric identities for cosine and sine of supplementary angles.

Step2: Find \(\cos(150^{\circ})\)

We know that \(\cos(180^{\circ}-\theta)=-\cos(\theta)\) (because cosine is negative in the second quadrant). Here \(\theta = 30^{\circ}\), so \(\cos(150^{\circ})=\cos(180^{\circ}- 30^{\circ})=-\cos(30^{\circ})\).
We know that \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\), so \(\cos(150^{\circ})=-\frac{\sqrt{3}}{2}\).

Step3: Find \(\sin(150^{\circ})\)

We know that \(\sin(180^{\circ}-\theta)=\sin(\theta)\) (because sine is positive in the second quadrant). Here \(\theta = 30^{\circ}\), so \(\sin(150^{\circ})=\sin(180^{\circ}- 30^{\circ})=\sin(30^{\circ})\).
We know that \(\sin(30^{\circ})=\frac{1}{2}\), so \(\sin(150^{\circ})=\frac{1}{2}\).

Answer:

\(\cos(150^{\circ})=-\frac{\sqrt{3}}{2}\), \(\sin(150^{\circ})=\frac{1}{2}\)