Question

use reference angles to find the exact value of the following expression. do not use a calculator. cos(15π/4)
determine the reference angle for 15π/4.
the reference angle is
(type your answer in radians. use integers or fractions for any numbers in the expression. type an exact answer, us)

Explanation:

Step1: Find the coterminal angle

First, find a coterminal angle of $\frac{15\pi}{4}$ that is between $0$ and $2\pi$. We know that $2\pi=\frac{8\pi}{4}$. Divide $\frac{15\pi}{4}$ by $2\pi$ (or $\frac{8\pi}{4}$) and find the remainder. $\frac{15\pi}{4}= 3\times\frac{8\pi}{4}-\frac{9\pi}{4}=3\times2\pi - \frac{9\pi}{4}$. Another way is to subtract $2\pi$ from $\frac{15\pi}{4}$ multiple - times. $\frac{15\pi}{4}-2\pi=\frac{15\pi - 8\pi}{4}=\frac{7\pi}{4}$.

Step2: Determine the reference angle

The angle $\theta=\frac{7\pi}{4}$ lies in the fourth - quadrant. The formula for the reference angle $\theta'$ of an angle $\theta$ in the fourth - quadrant is $\theta' = 2\pi-\theta$ (or $\theta'=| \theta - 2\pi|$). For $\theta=\frac{7\pi}{4}$, the reference angle $\theta'=2\pi-\frac{7\pi}{4}=\frac{8\pi - 7\pi}{4}=\frac{\pi}{4}$.

Step3: Find the cosine value

We know that $\cos\theta$ has the same absolute value as $\cos\theta'$ in the fourth - quadrant, and $\cos$ is positive in the fourth - quadrant. Since $\cos\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, then $\cos\frac{15\pi}{4}=\cos\frac{7\pi}{4}=\frac{\sqrt{2}}{2}$.

Answer:

The reference angle is $\frac{\pi}{4}$; $\cos\frac{15\pi}{4}=\frac{\sqrt{2}}{2}$