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{11}}{5}, \frac{\sqrt{14}}{5}\right)

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Answer:

Identify Point Quadrant: To find the angle $\theta$ that corresponds to a point on the unit circle, we can use the inverse trigonometric functions. The point $P$ has coordinates $(-\sqrt{11}/5, \sqrt{14}/5)$ . Since the $x$ -coordinate is negative and the $y$ -coordinate is positive, the point lies in the second quadrant.
Calculate Reference Angle: We can use the inverse tangent function to find the reference angle $\alpha$ , which is the acute angle to the x-axis. However, since the tangent function only gives us angles in the first and fourth quadrants, we need to adjust our approach. We can use the y-coordinate and x-coordinate to find the tangent of the reference angle $\alpha$ : $\tan(\alpha) = \left|\frac{y}{x}\right| = \left|\frac{\sqrt{14}/5}{-\sqrt{11}/5}\right| = \frac{\sqrt{14}}{\sqrt{11}}$ .
Find Reference Angle: Now we calculate the reference angle $\alpha$ using the inverse tangent function: $\alpha = \arctan(\sqrt{14}/\sqrt{11})$ . We use a calculator to find this angle.
Calculate Actual Angle: After calculating, we find that $\alpha \approx \arctan(\sqrt{14/11}) \approx 51.3$ degrees. This is the reference angle in the first quadrant.
Subtract Reference Angle: Since the point is in the second quadrant, we find the actual angle $\theta$ by subtracting the reference angle from $180$ degrees: $\theta = 180 - \alpha$ .
Subtract Reference Angle: Since the point is in the second quadrant, we find the actual angle $\theta$ by subtracting the reference angle from $180$ degrees: $\theta = 180 - \alpha$ . Subtracting the reference angle from $180$ degrees, we get $\theta \approx 180 - 51.3 \approx 128.7$ degrees.