Questions: Complete the sentence below. The point on the unit circle that corresponds to θ=π/4 is P=.

Transcript text: Complete the sentence below. The point on the unit circle that corresponds to $\theta=\frac{\pi}{4}$ is $P=$ $\qquad$ .

Solution

Solution Steps

Step 1: Recall the unit circle definition

The unit circle is defined as a circle with a radius of 1 centered at the origin $(0, 0)$ in the coordinate plane. Any angle $\theta$ measured from the positive $x$-axis corresponds to a point $P = (x, y)$ on the unit circle, where $x = \cos(\theta)$ and $y = \sin(\theta)$.

Step 2: Substitute $\theta = \frac{\pi}{4}$ into the trigonometric functions

For $\theta = \frac{\pi}{4}$, calculate $x$ and $y$: \[ x = \cos\left(\frac{\pi}{4}\right), \quad y = \sin\left(\frac{\pi}{4}\right). \]

Step 3: Evaluate $\cos\left(\frac{\pi}{4}\right)$ and $\sin\left(\frac{\pi}{4}\right)$

From trigonometric identities: \[ \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}, \quad \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}. \]

Step 4: Write the ordered pair $P$

Substitute the values of $x$ and $y$ into the ordered pair: \[ P = \left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right). \]