Find trigonometric ratios using the unit circle

Full solution

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}

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P=\left(-\frac{\sqrt{5}}{4}, \frac{\sqrt{11}}{4}\right)

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Answer:

Identify Coordinates: Identify the coordinates of point $P$ on the unit circle. $\newline$ The given point $P$ has coordinates $(-\sqrt{5}/4, \sqrt{11}/4)$ . Since the unit circle has a radius of $1$ , these coordinates correspond to $(\cos(\theta), \sin(\theta))$ for some angle $\theta$ in standard position.
Determine Quadrant: Determine the quadrant in which the angle $\theta$ lies. $\newline$ The $x$ -coordinate is negative and the $y$ -coordinate is positive, which means the point $P$ lies in the second quadrant.
Calculate Reference Angle: Calculate the reference angle $\theta'$ using the given coordinates. $\newline$ The reference angle $\theta'$ is found by taking the inverse cosine (arccos) of the absolute value of the x-coordinate of $P$ . However, since we are in the second quadrant, we need to use the y-coordinate to find the reference angle using the inverse sine (arcsin) function. $\newline$ $\theta' = \arcsin(\sqrt{11}/4)$
Actual Reference Angle: Calculate the actual value of the reference angle $\theta'$ . $\newline$ $\theta' = \arcsin(\sqrt{11}/4) \approx \arcsin(0.825)$ $\newline$ Using a calculator, we find: $\newline$ $\theta' \approx 55.7$ degrees
Determine Actual Angle: Determine the actual angle $\theta$ based on the reference angle $\theta'$ and the quadrant in which $\theta$ lies. $\newline$ Since $\theta$ is in the second quadrant, we calculate $\theta$ as: $\newline$ $\theta = 180 \text{ degrees} - \theta'$ $\newline$ $\theta = 180 \text{ degrees} - 55.7 \text{ degrees}$
Calculate Final Angle: Calculate the final value of $\theta$ . $\newline$ $\theta = 180 \text{ degrees} - 55.7 \text{ degrees} \approx 124.3 \text{ degrees}$