Algebra 2 Radians and the Unit Circle Worksheet (PDF)

Algebra 2 math worksheet: Radians and the Unit Circle – Maths PDF

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This unit circle worksheet for Algebra 2 covers radians and the unit circle (Common Core F-TF.A.1, F-TF.A.2, F-TF.A.3), with a free PDF to print.

These 20 problems practice radians and the unit circle (Common Core F-TF.A.1, F-TF.A.2 and F-TF.A.3). The first problems check converting between degrees and radians, the middle problems add exact trig values with reference angles and quadrant signs, and the last six are word problems and multi-step tasks that need arc length on wheels and wipers.

Work in order and show every step. Write each answer in simplest form, and open the answer key only after you have tried the problem yourself.

Algebra 2 · Common Core State Standards: F-TF.A.1, F-TF.A.2, F-TF.A.3

Unit circle worksheet: exercises 1 to 10

Exercise 1 – Degrees to radians

Convert \(60^{\circ}\) to radians.

Exercise 2 – A larger angle

Convert \(225^{\circ}\) to radians.

Exercise 3 – Radians to degrees

Convert \(\frac{7\pi}{6}\) to degrees.

Exercise 4 – Decimal radians

Convert 2 radians to degrees. Round to the nearest tenth.

Exercise 5 – Arc length

A sector: s = r times the central angle in radians.

A circle has radius 6 inches. Find the arc length for a central angle of \(\frac{\pi}{3}\).

Exercise 6 – Find the angle

An arc of length 15 feet lies on a circle of radius 10 feet. Find the central angle in radians.

Exercise 7 – Axis angle

Give the coordinates of the point on the unit circle at \(\frac{\pi}{2}\) and find \(\cos\frac{\pi}{2}\) and \(\sin\frac{\pi}{2}\).

Exercise 8 – Angle of 30 degrees

Quadrant I: the points (cos θ, sin θ) for the special angles.

Use the unit circle to evaluate \(\sin\frac{\pi}{6}\), \(\cos\frac{\pi}{6}\), and \(\tan\frac{\pi}{6}\).

Exercise 9 – Angle of 45 degrees

Evaluate \(\sin 45^{\circ}\), \(\cos 45^{\circ}\), and \(\tan 45^{\circ}\).

Exercise 10 – Quadrant II sine

Evaluate \(\sin\frac{2\pi}{3}\).

Exercises 11 and up

Exercise 11 – Quadrant II cosine

Signs by quadrant: All, Sine, Tangent, Cosine.

Evaluate \(\cos\frac{5\pi}{6}\).

Exercise 12 – Quadrant IV sine

Evaluate \(\sin\frac{7\pi}{4}\).

Exercise 13 – Tangent in quadrant II

Evaluate \(\tan\frac{3\pi}{4}\).

Exercise 14 – Straight angle

Evaluate \(\sin\pi\), \(\cos\pi\), and \(\tan\pi\).

Exercise 15 – Beyond one turn

Evaluate \(\sin\frac{13\pi}{6}\) by finding a coterminal angle.

Exercise 16 – Negative angle

Evaluate \(\cos\left(-\frac{\pi}{3}\right)\) and \(\sin\left(-\frac{\pi}{6}\right)\).

Exercise 17 – Solve sin θ

Angles around the unit circle in degrees and radians.

Find all angles \(\theta\) in \([0, 2\pi)\) with \(\sin\theta = \frac{\sqrt{2}}{2}\).

Exercise 18 – Solve cos θ

Find all angles \(\theta\) in \([0, 2\pi)\) with \(\cos\theta = -\frac{1}{2}\).

Exercise 19 – Identify the angle

The point \(\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\) is on the unit circle. Find the angle in \([0, 2\pi)\).

Exercise 20 – Ferris wheel

A sector: s = r times the central angle in radians.

A Ferris wheel has radius 30 feet. A rider turns through \(\frac{3\pi}{2}\) radians. How far does the rider travel?

Exercise 21 – Windshield wiper

A wiper blade 18 inches long sweeps an angle of \(120^{\circ}\). Find the arc length traced by the tip of the blade.

Exercise 22 – Find the error

Mei says \(\sin\frac{5\pi}{6} = -\frac{1}{2}\). Explain the error.

Key facts behind this unit circle worksheet

Key facts

θ = s / r, and 180° = π radians.
Arc length s = rθ when θ is in radians.
Point on the unit circle at θ: (cos θ, sin θ).
tan θ = sin θ / cos θ; cos² θ + sin² θ = 1.
Use reference angles and quadrant signs for exact values.

How to use this unit circle worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the unit circle worksheet as a PDF with the button below and keep it with your notes.

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