Question

Given Cos(30°)= sqrt3/2 and Sin(30°)=1/2, determine the following in exact form: cos(150°)+ sin(210°)

Answer 1

Answer:

-(√3 + 1)/2

Explanation:

We will use the following properties:

$\begin{gathered} \cos (A-B)=\cos (A)\cos (B)+\sin (A)\sin (B) \\ \sin (A+B)=\sin (A)\cos (B)+\cos (A)\sin (B) \end{gathered}$

First, let's calculate cos(150). Since 150 = 180 - 30, we get:

$\cos (180-30)=\cos (180)\cos (30)+\sin (180)\sin (30)$

Taking into account that cos(180) = -1 and sin(180) = 0, we get:

$\begin{gathered} \cos (150)=-1\cdot\frac{\sqrt[]{3}}{2}+0\cdot\frac{1}{2} \\ \cos (150)=-\frac{\sqrt[]{3}}{2} \end{gathered}$

On the other hand, 210 = 180 + 30, so sin(210) will be equal to:

$\begin{gathered} \sin (180+30)=\sin (180)\cos (30)+\cos (180)\sin (30) \\ \sin (210)=0\cdot\frac{\sqrt[]{3}}{2}+(-1)\cdot\frac{1}{2} \\ \sin (210)=-\frac{1}{2} \end{gathered}$

Therefore, cos (150) + sin(210) is equal to:

$\cos (150)+\sin (210)=-\frac{\sqrt[]{3}}{2}-\frac{1}{2}=-\frac{(\sqrt[]{3}+1)}{2}$

So, the answer is: -(√3 + 1)/2