Question

Determine the exact value of cos 135 degrees and explain how you knew to use the side lengths you used

Answer 1

Explanation

We are required to determine the exact value of cos 135°.

Since the angle lies in the second quadrant, we have:

$\begin{gathered} \cos135\degree=\cos(180\degree-45\degree) \\ \cos135\degree=-\cos45\degree \end{gathered}$

To determine the value of x, we have:

$\begin{gathered} \text{ Using the Pythagorean theorem,} \\ x^2=1^2+1^2 \\ x=\sqrt{1^2+1^2} \\ x=\sqrt{1+1} \\ x=\sqrt{2} \end{gathered}$

Therefore, the value of cos 135° is:

$\begin{gathered} \text{ We know that }cos\theta=\frac{adj}{hyp} \\ \therefore\cos135\degree=-\cos45\degree=-\frac{1}{\sqrt{2}} \\ \cos135\degree=-\frac{1}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}} \\ \cos135\degree=-\frac{\sqrt{2}}{2} \end{gathered}$

Hence, the answer is:

$\cos(135)\operatorname{\degree}=-\frac{\sqrt{2}}{2}$

The lengths used is the lowest length of sides that can be used.