Find function values for the sine and cosine of the special angles.
Use reference angles to evaluate trigonometric functions.
Evaluate sine and cosine values using a calculator.
Satellite Orbit Analysis
A communications satellite orbits Earth in a circular path. Engineers track the satellite’s position relative to a ground station at the center of its orbit. The satellite’s path can be modeled using a unit circle, where each point [latex](x, y)[/latex] corresponds to the satellite’s position at angle [latex]t[/latex].
On a unit circle with radius 1, for any angle [latex]t[/latex], the coordinates of the point where the terminal side intersects the circle are [latex](\cos t, \sin t)[/latex]. The cosine gives the x-coordinate and the sine gives the y-coordinate.
How To: Finding Sine and Cosine Using Reference Angles
Determine which quadrant the angle is in
Find the reference angle (the acute angle to the x-axis)
Find sine and cosine of the reference angle
Apply appropriate signs based on the quadrant (remember: All Students Take Calculus)
Quadrant I: both positive
Quadrant II: sine positive, cosine negative
Quadrant III: both negative
Quadrant IV: sine negative, cosine positive
A satellite begins its orbit at the rightmost position (angle 0) relative to a ground tracking station. Engineers need to determine the satellite’s position at various points in its orbit.
After rotating [latex]\frac{5\pi}{6}[/latex] radians, find the coordinates on a unit circle corresponding to this angle.
The satellite’s actual orbit has a radius of 8,000 kilometers above the tracking station. What are the satellite’s horizontal and vertical distances from the station?
Find the satellite’s position when it has rotated [latex]\frac{5\pi}{4}[/latex] radians from its starting point.
First, determine the quadrant. Since [latex]\frac{5\pi}{6}[/latex] is between [latex]\frac{\pi}{2}[/latex] and [latex]\pi[/latex], the angle is in Quadrant II.
For the reference angle [latex]\frac{\pi}{6}[/latex] (30°), we know from our special angles: [latex]\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \text{ and } \sin\left(\frac{\pi}{6}\right) = \frac{1}{2}[/latex]
In Quadrant II, cosine is negative and sine is positive: [latex]\cos\left(\frac{5\pi}{6}\right) = -\frac{\sqrt{3}}{2} \text{ and } \sin\left(\frac{5\pi}{6}\right) = \frac{1}{2}[/latex]
The unit circle coordinates are [latex]\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)[/latex].
For a circular orbit with radius 8,000 kilometers, multiply the unit circle coordinates by 8,000:
Horizontal distance from station: [latex]8{,}000 \times \left(-\frac{\sqrt{3}}{2}\right) \approx 8{,}000 \times (-0.866) = -6{,}928[/latex] km
Vertical distance from station: [latex]8{,}000 \times \frac{1}{2} = 4{,}000[/latex] km
The satellite is approximately 6,928 km to the west of the tracking station and 4,000 km north of the station.
First, determine the quadrant. Since [latex]\frac{5\pi}{4}[/latex] is between [latex]\pi[/latex] and [latex]\frac{3\pi}{2}[/latex], the angle is in Quadrant III.
For the reference angle [latex]\frac{\pi}{4}[/latex] (45°): [latex]\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \text{ and } \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}[/latex]
In Quadrant III, both cosine and sine are negative: [latex]\cos\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2} \text{ and } \sin\left(\frac{5\pi}{4}\right) = -\frac{\sqrt{2}}{2}[/latex]
The unit circle coordinates are [latex]\left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)[/latex].
On the 8,000 km orbit:
Horizontal: [latex]8{,}000 \times \left(-\frac{\sqrt{2}}{2}\right) \approx -5{,}657[/latex] km (west of station)
Vertical: [latex]8{,}000 \times \left(-\frac{\sqrt{2}}{2}\right) \approx -5{,}657[/latex] km (south of station)
The satellite is approximately 5,657 km southwest of the tracking station.
To find coordinates on any circle with radius [latex]r[/latex], multiply the unit circle coordinates by [latex]r[/latex]: [latex](x, y) = (r\cos t, r\sin t)[/latex].
A satellite is at an angle of [latex]\frac{2\pi}{3}[/latex] radians from its starting position.
Find the unit circle coordinates for this angle.
If the satellite orbits at 6,500 km from the tracking station, what are its horizontal and vertical distances from the station?