using only the unit circle definition of the sine function, find the solution set of the equation on the…
using only the unit circle definition of the sine function, find the solution set of the equation on the given interval. |sin x| = √2/2 on 0,2π select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the solution set is { }. (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression. b. the solution set is the interval . (type your answer in interval notation. simplify your answer. type an exact answer, using π as needed. use integers or fractio c. there is no solution.
Answer
Explanation:
Step1: Recall unit - circle sine values
On the unit - circle, $\sin x$ represents the $y$-coordinate of the point on the unit - circle corresponding to the angle $x$. We know that $\sin x=\frac{\sqrt{2}}{2}$ or $\sin x =-\frac{\sqrt{2}}{2}$ since $|\sin x|=\frac{\sqrt{2}}{2}$.
Step2: Find angles for $\sin x=\frac{\sqrt{2}}{2}$
On the interval $[0,2\pi]$, $\sin x=\frac{\sqrt{2}}{2}$ when $x = \frac{\pi}{4}$ and $x=\frac{3\pi}{4}$ because at these angles, the $y$-coordinate of the points on the unit - circle is $\frac{\sqrt{2}}{2}$.
Step3: Find angles for $\sin x=-\frac{\sqrt{2}}{2}$
On the interval $[0,2\pi]$, $\sin x=-\frac{\sqrt{2}}{2}$ when $x=\frac{5\pi}{4}$ and $x = \frac{7\pi}{4}$ since the $y$-coordinate of the points on the unit - circle is $-\frac{\sqrt{2}}{2}$ at these angles.
Answer:
A. The solution set is $\left{\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}\right}$