Question

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

Answer 1

-(√3 + 1)/2

Step-by-step 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(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(1)/(2) \\ \sin (210)=-(1)/(2) \end{gathered}`

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

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

So, the answer is: -(√3 + 1)/2