Question

Use an Addition orSubtraction Formula to find the exact value of the expression, asDemonstrated$\mathbf{cos}\mathbf{}{\mathbf{195}}^{\mathbf{\circ }}$

Short answer

Using subtraction formula of sine. We get$\mathbf{cos}\mathbf{}{\mathbf{195}}^{\mathbf{\circ }}\mathbf{=}\mathbf{-}\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)$

Step-by-step solution

Step 1. Given information.

We know that sine and cosine are the trigonometric functions, Where x and y

are the angles

If we know the values of $\sin x$, $\sin y$, $\cos x$, $\cos y$, then by using

subtraction formula of sine we can easily calculate the value of $\cos {195}^{\circ }$.

Formula$\cos (x-y)=\cos x\cos y+\sin x\sin y$

Step 2. Determining the value.

Split the angle of $\cos {195}^{\circ }$,

That is ${195}^{\circ }={180}^{\circ }+{15}^{\circ }$, Since we know the exact values of sine and cosine at 45 and 30

So, we got the values of x and y, $x={180}^{\circ }$

And $y={15}^{\circ }$

Now apply the addition formula of sine to get $\cos {195}^{\circ }$
By using $\cos (x-y)=\cos x\cos y+\sin x\sin y$

$\cos \left({195}^{\circ }\right)=-\{\cos {60}^{\circ }\cos {45}^{\circ }+\sin {60}^{\circ }\sin {45}^{\circ }\}$

$\cos \left({195}^{\circ }\right)=-\left(\frac{1}{2\sqrt{2}}+\frac{\sqrt{3}}{2\sqrt{2}}\right)$

$\cos \left({105}^{\circ }\right)=-\left(\frac{1+\sqrt{3}}{2\sqrt{2}}\right)$