Given the following point on the unit circle, find the angle, to the nearest tenth of a degree (if necessary), of the terminal side through that point, $0^{\circ} \leq \theta<360^{\circ}$ . $\newline$ $P=\left(\frac{3}{5}, \frac{4}{5}\right)$ $\newline$ Answer:

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Algebra 2

Find trigonometric ratios using the unit circle

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Q. Given the following point on the unit circle, find the angle, to the nearest tenth of a degree (if necessary), of the terminal side through that point,

0^{\circ} \leq \theta<360^{\circ}

\newline

P=\left(\frac{3}{5}, \frac{4}{5}\right)

\newline

Answer:

Identify Coordinates: Identify the coordinates of point $P$ on the unit circle. $\newline$ The given point $P$ has coordinates $(x, y) = (\frac{3}{5}, \frac{4}{5})$ . On the unit circle, these coordinates correspond to $(\cos(\theta), \sin(\theta))$ .
Determine Reference Angle: Determine the reference angle. $\newline$ Since both $x$ and $y$ coordinates are positive, point $P$ lies in the first quadrant. The reference angle is the angle $\theta'$ that the terminal side makes with the $x$ -axis. To find $\theta'$ , we use the $\text{arccosine}$ or $\text{arcsine}$ function. However, since we are in the first quadrant, $\theta'$ is simply $\theta$ .
Calculate Using Arccosine: Calculate the angle using the arccosine function. $\newline$ We use the x-coordinate ( $\cos(\theta)$ ) to find the angle $\theta$ . Thus, $\theta = \arccos(\frac{3}{5})$ . We calculate this using a calculator.
Convert to Degrees: Convert the angle from radians to degrees. $\newline$ After calculating the arccosine, we get $\theta$ in radians. To convert it to degrees, we multiply by $\frac{180}{\pi}$ . However, since we are using a calculator, it can directly give us the angle in degrees.
Round to Nearest Tenth: Round the angle to the nearest tenth of a degree. $\newline$ Using a calculator, we find that $\theta \approx \arccos(\frac{3}{5}) \approx 53.13$ degrees. Rounding to the nearest tenth of a degree, we get $\theta \approx 53.1$ degrees.