explain why $cos(-\frac{3pi}{4})=cos(\frac{5pi}{4})$. you can choose to type, draw, or use both to share…

Explanation:

Step1: Recall cosine function property

The cosine function is an even - function, i.e., $\cos(-x)=\cos(x)$. So, $\cos(-\frac{3\pi}{4})=\cos(\frac{3\pi}{4})$.

Step2: Use angle - addition formula for cosine

We know that $\cos(x + \pi)=-\cos(x)$. Also, $\frac{5\pi}{4}=\frac{\pi}{4}+\pi$ and $\frac{3\pi}{4}=\pi-\frac{\pi}{4}$. Another way is to consider the unit - circle. The angle $\frac{3\pi}{4}$ and $\frac{5\pi}{4}$ have the same $x$ - coordinate of the corresponding points on the unit circle. The reference angle for $\frac{3\pi}{4}$ is $\pi-\frac{3\pi}{4}=\frac{\pi}{4}$, and for $\frac{5\pi}{4}$ is $\frac{5\pi}{4}-\pi=\frac{\pi}{4}$. In the unit - circle, the cosine of an angle is the $x$ - coordinate of the point on the unit circle corresponding to that angle. The angles $\frac{3\pi}{4}$ and $\frac{5\pi}{4}$ are symmetric about the $x$ - axis in a sense related to the unit - circle, and $\cos(\frac{3\pi}{4})=\cos(\frac{5\pi}{4})$. Since $\cos(-\frac{3\pi}{4})=\cos(\frac{3\pi}{4})$, we have $\cos(-\frac{3\pi}{4})=\cos(\frac{5\pi}{4})$.

Answer:

The cosine function is even, so $\cos(-\frac{3\pi}{4})=\cos(\frac{3\pi}{4})$. Also, considering the unit - circle, the angles $\frac{3\pi}{4}$ and $\frac{5\pi}{4}$ have the same $x$ - coordinate of the corresponding points on the unit circle, so $\cos(\frac{3\pi}{4})=\cos(\frac{5\pi}{4})$. Thus, $\cos(-\frac{3\pi}{4})=\cos(\frac{5\pi}{4})$.