Question

Use an Addition orSubtraction Formula to find the exact value of the expression, asDemonstrated$\mathbf{cos}{\mathbf{105}}^{\mathbf{\circ }}$

Short answer

Using subtraction formula of sine. We get$\mathbf{cos}{\mathbf{105}}^{\mathbf{\circ }}\mathbf{=}\frac{\sqrt{\mathbf{2}}\mathbf{-}\sqrt{\mathbf{6}}}{\mathbf{4}}$

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 $\cos {105}^{\circ }$.

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

Step 2. Determining the value.

Split the angle of $\cos {105}^{\circ }$,

That is ${105}^{\circ }={60}^{\circ }+{45}^{\circ }$, Since we know the exact values of sine and cosine at 45 and 30

So, we got the values of x and y, $x={60}^{\circ }$

And $y={45}^{\circ }$

Now apply the addition formula of sine to get $\cos {105}^{\circ }$

By using $\cos (x+y)=\cos x\cos y-\sin x\sin y$

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

$\cos ({60}^{\circ }+{45}^{\circ })=\frac{\sqrt{2}-\sqrt{3}\sqrt{2}}{4}$

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

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

$\cos \left({105}^{\circ }\right)=\frac{\sqrt{2}-\sqrt{6}}{4}$