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].