Radians and the Unit Circle: Algebra 2 Math Worksheet – PDF Download

Problem 1: Degrees to radians

Convert each angle to radians. Leave the answer in terms of \(\pi\).

1. \(45^{\circ}\)
2. \(120^{\circ}\)
3. \(270^{\circ}\)
4. \(15^{\circ}\)

Problem 2: Radians to degrees

Convert each angle to degrees.

1. \(\frac{\pi}{6}\)
2. \(\frac{5\pi}{4}\)
3. \(\frac{7\pi}{3}\)
4. \(-\frac{\pi}{2}\)

Problem 3: Arc length

Find the arc length for each central angle.

1. Radius 6 inches and central angle \(\frac{\pi}{3}\). Give the exact length and a decimal rounded to hundredths.
2. Radius 4 feet and central angle 2.5 radians.

Problem 4: Angle from an arc

An arc of length 9 inches lies on a circle of radius 6 inches. Find the central angle in radians and in degrees to the nearest tenth.

Problem 5: Locating the quadrant

Name the quadrant of the terminal side of each angle.

1. \(\frac{5\pi}{6}\) (point A)
2. \(\frac{4\pi}{3}\) (point B)
3. \(\frac{7\pi}{4}\) (point C)
4. \(\frac{\pi}{3}\) (point D)

Problem 6: Special angle values

Write the exact value of each expression.

1. \(\sin\frac{\pi}{6}\)
2. \(\cos\frac{\pi}{4}\)
3. \(\tan\frac{\pi}{3}\)
4. \(\cos\frac{\pi}{3}\)

Problem 7: Unit circle coordinates

Find the exact coordinates of the point where the terminal side of each angle meets the unit circle.

1. \(\frac{2\pi}{3}\)
2. \(\frac{5\pi}{4}\)
3. \(\frac{11\pi}{6}\)

Problem 8: Reference angles

Find the reference angle for each angle.

1. \(\frac{5\pi}{6}\)
2. \(\frac{7\pi}{6}\)
3. \(\frac{5\pi}{3}\)
4. \(\frac{3\pi}{4}\)

Problem 9: Quadrantal angles

Complete the table of exact values. Write undefined where it applies.

\(\theta\)	\(\sin\theta\)	\(\cos\theta\)	\(\tan\theta\)
\(0\)
\(\frac{\pi}{2}\)
\(\pi\)
\(\frac{3\pi}{2}\)

Problem 10: Evaluating in other quadrants

Find the exact value.

1. \(\sin(\frac{7\pi}{6})\)
2. \(\cos(\frac{5\pi}{3})\)
3. \(\tan(\frac{3\pi}{4})\)

Problem 11: Negative and large angles

Find the exact value.

1. \(\cos(-\frac{\pi}{3})\)
2. \(\sin(-\frac{\pi}{4})\)
3. \(\sin\frac{13\pi}{6}\)
4. \(\cos\frac{9\pi}{4}\)

Problem 12: Solving for the angle

Find all angles \(\theta\) with \(0\le\theta< 2\pi\) that satisfy each equation.

1. \(\sin\theta=\frac{\sqrt{3}}{2}\)
2. \(\cos\theta=-\frac{\sqrt{2}}{2}\)

Problem 13: Checking the identity

Verify that \(\sin^{2}\theta+\cos^{2}\theta=1\) for each angle by substituting exact values.

1. \(\theta=\frac{\pi}{6}\)
2. \(\theta=\frac{5\pi}{4}\)

Problem 14: Cosine from sine

Given \(\sin\theta=\frac{3}{5}\) with the terminal side of \(\theta\) in Quadrant II, find \(\cos\theta\) and \(\tan\theta\).

Problem 15: Sine from cosine

Given \(\cos\theta=\frac{5}{13}\) with the terminal side in Quadrant IV, find \(\sin\theta\) and \(\tan\theta\).

Problem 16: Using the tangent

Given \(\tan\theta=\frac{4}{3}\) with the terminal side in Quadrant III, find \(\sin\theta\) and \(\cos\theta\).

Problem 17: Radical values

Given \(\cos\theta=-\frac{\sqrt{5}}{3}\) with the terminal side in Quadrant III, find \(\sin\theta\) and \(\tan\theta\).

Problem 18: Ferris wheel height

A Ferris wheel has radius 25 feet and its center is 30 feet above the ground. Mei’s height is \(h=30+25\sin\theta\), where \(\theta\) is measured from the horizontal through the center.

Find Mei’s height for each angle. Round to the nearest tenth when needed.

1. \(\theta=\frac{\pi}{6}\)
2. \(\theta=\frac{\pi}{2}\)
3. \(\theta=\frac{5\pi}{4}\)
4. \(\theta=\frac{3\pi}{2}\)

Problem 19: Clock hand

Sofia’s wall clock has a minute hand 5 inches long. In 20 minutes the hand turns through \(\frac{1}{3}\) of a full circle.

1. Write the angle in radians.
2. How far does the tip of the minute hand travel? Give the exact value and a decimal to the nearest hundredth.

Problem 20: Pizza slice

Liam cuts a pizza of radius 8 inches into slices with a central angle of \(45^{\circ}\).

1. Convert \(45^{\circ}\) to radians.
2. Find the length of the crust on one slice to the nearest hundredth of an inch.

Problem 21: Bicycle wheel

A bicycle wheel has radius 13 inches. Noah rides until the wheel turns exactly 3 full revolutions.

1. Find the angle the wheel turns through, in radians.
2. Find the distance he travels in inches and in feet, to the nearest hundredth.

Problem 22: Point on the circle

The point \(P(-\frac{12}{13},\ y)\) lies on the unit circle in Quadrant III, and it is on the terminal side of angle \(\theta\). Find \(y\), then \(\sin\theta\), \(\cos\theta\) and \(\tan\theta\).

Problem 23: When is the rider low

On the same wheel, \(h=30+25\sin\theta\). Find all angles \(\theta\) with \(0\le\theta< 2\pi\) for which Mei is 17.5 feet above the ground.

See every chapter on the Algebra 2 math page.