Key takeaway: This precalculus unit circle lesson explains how angles in radians connect to points \((\cos\theta,\sin\theta)\) on a circle of radius 1. A radian measures the angle that cuts off an arc as long as the radius, so \(\pi\) radians equals \(180^{\circ}\). Reference angles and symmetry then turn three special angles into every exact value. The page supports standard F-TF.A.2, and it closes with arc length and angular speed.
A radian is the angle whose arc equals the radius, so pi radians equals 180 degrees. On the unit circle the point at angle theta is (cos theta, sin theta). Reference angles and symmetry turn the values at pi/6, pi/4 and pi/3 into every other angle, and arc length is s = r times theta.
Precalculus Unit Circle Lesson: key facts
- 180 degrees = pi radians; multiply degrees by pi/180 to get radians.
- The point at angle theta on the unit circle is (cos theta, sin theta), and tan theta = sin theta / cos theta.
- Special values: sin and cos of pi/6, pi/4, pi/3 are 1/2, sqrt2/2, sqrt3/2 in some order.
- Arc length is s = r*theta and linear speed is v = r*omega, with theta in radians.
- Sine and cosine have period 2pi, tangent has period pi; cosine is even, sine and tangent are odd.
What is a radian?
A radian is the measure of a central angle whose arc is exactly as long as the radius. A full circle has circumference \(2\pi r\), so one full turn is \(2\pi\) radians and \(\pi\) radians equals \(180^{\circ}\). To convert, multiply degrees by \(\frac{\pi}{180}\), or radians by \(\frac{180}{\pi}\). For example, \(135^{\circ}=\frac{3\pi}{4}\).
Unit circle: the circle \(x^{2}+y^{2}=1\) centered at the origin. The point at angle \(\theta\) is \((\cos\theta,\sin\theta)\).
Coterminal angles: angles that share a terminal side, such as \(\theta\) and \(\theta+2\pi\). Reference angle: the acute angle between the terminal side and the x-axis.
What are the special angle values on the unit circle?
Because the point at angle \(\theta\) has coordinates \((\cos\theta,\sin\theta)\), every value comes from reading a coordinate, and \(\tan\theta=\frac{\sin\theta}{\cos\theta}\).
| Angle | \(\cos\theta\) | \(\sin\theta\) | \(\tan\theta\) |
|---|---|---|---|
| \(\frac{\pi}{6}\) (30°) | \(\frac{\sqrt{3}}{2}\) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{3}\) |
| \(\frac{\pi}{4}\) (45°) | \(\frac{\sqrt{2}}{2}\) | \(\frac{\sqrt{2}}{2}\) | \(1\) |
| \(\frac{\pi}{3}\) (60°) | \(\frac{1}{2}\) | \(\frac{\sqrt{3}}{2}\) | \(\sqrt{3}\) |
How do reference angles and symmetry give every other value?
- Rewrite the angle between \(0\) and \(2\pi\) by adding or subtracting full turns.
- Find the reference angle: \(\pi-\theta\) in Quadrant II, \(\theta-\pi\) in Quadrant III, \(2\pi-\theta\) in Quadrant IV.
- Read the value for the reference angle from the table.
- Attach the sign: sine is positive in Quadrants I and II, cosine in I and IV, tangent in I and III.
Example. For \(\sin\frac{5\pi}{6}\), the reference angle is \(\pi-\frac{5\pi}{6}=\frac{\pi}{6}\) and sine is positive in Quadrant II, so \(\sin\frac{5\pi}{6}=\frac{1}{2}\).
How do arc length and angular speed work?
On a circle of radius \(r\), a central angle of \(\theta\) radians cuts off an arc of length \(s=r\theta\). If the angle changes at \(\omega=\frac{\theta}{t}\) radians per unit time, the linear speed is \(v=r\omega\). With \(r=10\) in and \(\theta=\frac{2\pi}{3}\), the arc length is \(s=\frac{20\pi}{3}\approx 20.94\) in.
Why are the trigonometric functions periodic, even or odd?
Adding a full turn returns to the same point, so \(\sin(\theta+2\pi)=\sin\theta\) and \(\cos(\theta+2\pi)=\cos\theta\); tangent repeats every \(\pi\). Reflecting across the x-axis gives \(\cos(-\theta)=\cos\theta\) (even) and \(\sin(-\theta)=-\sin\theta\), \(\tan(-\theta)=-\tan\theta\) (odd).
Angles, Radians and the Unit Circle: frequently asked questions
Why do mathematicians use radians instead of degrees?
Radians connect angle directly to length: an angle of theta radians on a circle of radius r cuts off an arc of length r times theta. This makes formulas such as arc length and angular speed simple, with no conversion factor.
How do you remember the unit circle values?
Memorize the first quadrant: the angles pi/6, pi/4 and pi/3 have sine values 1/2, sqrt2/2, sqrt3/2 and cosine values in the reverse order. Reference angles and signs then give every other angle.
What is a reference angle?
It is the acute angle between the terminal side of an angle and the x-axis. The trigonometric values of an angle equal those of its reference angle, up to a sign that depends on the quadrant.
What does coterminal mean?
Two angles are coterminal if their terminal sides coincide, so they differ by a whole number of full turns. For instance, 7pi/6 and -5pi/6 are coterminal, and they have the same sine, cosine and tangent.
How do you convert radians per minute into linear speed?
Multiply the angular speed in radians per unit time by the radius: v = r times omega. A wheel of radius 13 inches turning at 6pi radians per second moves a point on its rim at 78pi inches per second.
What is a radian, and how do you convert to it?
Multiply degrees by \(\frac{\pi}{180}\). For example, \(240^{\circ}\) becomes \(\frac{4\pi}{3}\), and \(135^{\circ}\) becomes \(\frac{3\pi}{4}\). Conversely, multiply radians by \(\frac{180}{\pi}\). Keep \(\pi\) in the answer, because it gives an exact value.
How do reference angles give exact values?
Take \(\frac{5\pi}{4}\). It sits in Quadrant III, and its reference angle is \(\frac{\pi}{4}\). Both coordinates are negative there, so the point is \((-\frac{\sqrt{2}}{2},-\frac{\sqrt{2}}{2})\). First find the reference angle, then attach the signs of the quadrant.
How do you use the precalculus unit circle lesson for arc length?
The formula is \(s=r\theta\), and the angle must be in radians. A sector with radius 8 ft and a \(135^{\circ}\) angle uses \(\theta=\frac{3\pi}{4}\), so \(s=6\pi\) ft. Using 135 directly would give a wrong, enormous number.
Why are sine and cosine periodic, even or odd?
Adding \(2\pi\) returns to the same point, so both functions repeat. Reflecting across the \(x\)-axis flips only the \(y\)-coordinate. As a result, \(\cos(-\theta)=\cos\theta\) and \(\sin(-\theta)=-\sin\theta\).
Where can you take this circle knowledge next?
Draw a blank unit circle from memory and fill in the coordinates for every multiple of \(\frac{\pi}{6}\) and \(\frac{\pi}{4}\). Afterward, time yourself on ten conversions. Finally, move to the worksheet, where the same ideas power wheel and clock problems.
See every chapter on the Precalculus math page.
