write the expression as a function of \\( \\theta \\), with no angle measure involved. \\( \\cos…

Explanation:

Step1: Use the cosine addition formula

The cosine addition formula is $\cos(A + B)=\cos A\cos B-\sin A\sin B$. Here $A = 150^{\circ}$ and $B=\theta$. So, $\cos(150^{\circ}+\theta)=\cos150^{\circ}\cos\theta-\sin150^{\circ}\sin\theta$.

Step2: Find the values of $\cos150^{\circ}$ and $\sin150^{\circ}$

We know that $\cos150^{\circ}=\cos(180^{\circ} - 30^{\circ})=-\cos30^{\circ}=-\frac{\sqrt{3}}{2}$ and $\sin150^{\circ}=\sin(180^{\circ}-30^{\circ})=\sin30^{\circ}=\frac{1}{2}$.

Step3: Substitute the values into the formula

Substitute $\cos150^{\circ}=-\frac{\sqrt{3}}{2}$ and $\sin150^{\circ}=\frac{1}{2}$ into $\cos150^{\circ}\cos\theta-\sin150^{\circ}\sin\theta$. We get $\left(-\frac{\sqrt{3}}{2}\right)\cos\theta-\frac{1}{2}\sin\theta=-\frac{\sqrt{3}\cos\theta+\sin\theta}{2}$.

Answer:

$-\frac{\sqrt{3}\cos\theta+\sin\theta}{2}$