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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^(@) <= theta < 360^(@). P=(-(sqrt5)/(4),(sqrt11)/(4)) Answer:

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} .\newlineP=(−54,114) P=\left(-\frac{\sqrt{5}}{4}, \frac{\sqrt{11}}{4}\right) \newlineAnswer:

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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∘≤θ<360∘ 0^{\circ} \leq \theta<360^{\circ} .\newlineP=(−54,114) P=\left(-\frac{\sqrt{5}}{4}, \frac{\sqrt{11}}{4}\right) \newlineAnswer:
  1. Identify Coordinates: Identify the coordinates of point PP on the unit circle.\newlineThe given point PP has coordinates (−5/4,11/4)(-\sqrt{5}/4, \sqrt{11}/4). Since the unit circle has a radius of 11, these coordinates correspond to (cos⁡(θ),sin⁡(θ))(\cos(\theta), \sin(\theta)) for some angle θ\theta in standard position.
  2. Determine Quadrant: Determine the quadrant in which the angle θ\theta lies.\newlineThe xx-coordinate is negative and the yy-coordinate is positive, which means the point PP lies in the second quadrant.
  3. Calculate Reference Angle: Calculate the reference angle θ′\theta' using the given coordinates.\newlineThe reference angle θ′\theta' is found by taking the inverse cosine (arccos) of the absolute value of the x-coordinate of PP. 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.\newlineθ′=arcsin⁡(11/4)\theta' = \arcsin(\sqrt{11}/4)
  4. Actual Reference Angle: Calculate the actual value of the reference angle θ′\theta'.\newlineθ′=arcsin⁡(11/4)≈arcsin⁡(0.825)\theta' = \arcsin(\sqrt{11}/4) \approx \arcsin(0.825)\newlineUsing a calculator, we find:\newlineθ′≈55.7\theta' \approx 55.7 degrees
  5. Determine Actual Angle: Determine the actual angle θ\theta based on the reference angle θ′\theta' and the quadrant in which θ\theta lies.\newlineSince θ\theta is in the second quadrant, we calculate θ\theta as:\newlineθ=180 degrees−θ′\theta = 180 \text{ degrees} - \theta'\newlineθ=180 degrees−55.7 degrees\theta = 180 \text{ degrees} - 55.7 \text{ degrees}
  6. Calculate Final Angle: Calculate the final value of θ\theta.\newlineθ=180 degrees−55.7 degrees≈124.3 degrees\theta = 180 \text{ degrees} - 55.7 \text{ degrees} \approx 124.3 \text{ degrees}

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