Trigonometry Practice

Unit Circle Practice Worksheet

Practice angles, radians, coordinates, reference angles, and exact sine, cosine, and tangent values.

Start Practicing Review the Unit Circle Guide

Try each problem on your own first. Then click Show solution to check the conversion, reference angle, exact value, and reasoning.

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Unit Circle Essentials

Coordinates

\[ (x,y) = (\cos\theta,\sin\theta) \]

Tangent

\[ \tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x} \]

Angle Conversion

\[ 180^\circ=\pi \text{ radians} \]

Quadrant Signs

I: all, II: sine, III: tangent, IV: cosine

Download Practice PDF Open Unit Circle Reference

Level 1

Degrees and Radians

Convert between angle measures and identify equivalent degree and radian values.

Problem 1

Degrees to Radians

Convert the angle to radians.

\[ 135^\circ \]

Step-by-Step Solution

Multiply the degree measure by \(\dfrac{\pi}{180^\circ}\).

\[ \begin{aligned} 135^\circ \times \frac{\pi}{180^\circ} &= \frac{135\pi}{180} \\[6pt] &= \frac{3\pi}{4} \end{aligned} \]

The degree symbols cancel, and the fraction \(\dfrac{135}{180}\) simplifies to \(\dfrac{3}{4}\).

Answer: \[ 135^\circ = \frac{3\pi}{4} \]

Problem 2

Radians to Degrees

Convert the angle to degrees.

\[ \frac{5\pi}{6} \]

Step-by-Step Solution

Multiply the radian measure by \(\dfrac{180^\circ}{\pi}\).

\[ \begin{aligned} \frac{5\pi}{6} \times \frac{180^\circ}{\pi} &= \frac{5(180^\circ)}{6} \\[6pt] &= 5(30^\circ) \\[6pt] &= 150^\circ \end{aligned} \]

The \(\pi\) factors cancel before multiplying.

Answer: \[ \frac{5\pi}{6} = 150^\circ \]

Problem 3

Coterminal Angles

Find one positive angle coterminal with the given angle.

\[ -\frac{\pi}{3} \]

Step-by-Step Solution

Add one complete rotation, \(2\pi\), to the negative angle.

\[ \begin{aligned} -\frac{\pi}{3}+2\pi &= -\frac{\pi}{3} + \frac{6\pi}{3} \\[6pt] &= \frac{5\pi}{3} \end{aligned} \]

The angles \(-\dfrac{\pi}{3}\) and \(\dfrac{5\pi}{3}\) end at the same position on the unit circle.

Answer: \[ \frac{5\pi}{3} \]

Level 2

Coordinates and Exact Values

Use unit-circle coordinates to determine exact sine, cosine, and tangent values.

Problem 4

Unit Circle Coordinates

Find the point on the unit circle corresponding to the angle.

\[ \theta=\frac{5\pi}{6} \]

Step-by-Step Solution

The angle \(\dfrac{5\pi}{6}\) is \(150^\circ\), which lies in Quadrant II.

Its reference angle is \(\dfrac{\pi}{6}\). The first-quadrant coordinates for \(\dfrac{\pi}{6}\) are

\[ \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right). \]

In Quadrant II, the \(x\)-coordinate is negative and the \(y\)-coordinate is positive.

Answer: \[ \left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right) \]

Solution Graph

Problem 5

Exact Trigonometric Values

Find the exact values of sine, cosine, and tangent.

\[ \theta=\frac{7\pi}{4} \]

Step-by-Step Solution

The angle \(\dfrac{7\pi}{4}\) is \(315^\circ\), which lies in Quadrant IV. Its reference angle is \(\dfrac{\pi}{4}\).

The coordinate magnitudes for \(\dfrac{\pi}{4}\) are both \(\dfrac{\sqrt{2}}{2}\). In Quadrant IV, cosine is positive and sine is negative.

\[ \begin{aligned} \cos\left(\frac{7\pi}{4}\right) &= \frac{\sqrt{2}}{2} \\[6pt] \sin\left(\frac{7\pi}{4}\right) &= -\frac{\sqrt{2}}{2} \end{aligned} \]

Divide sine by cosine to find tangent.

\[ \begin{aligned} \tan\left(\frac{7\pi}{4}\right) &= \frac{ -\frac{\sqrt{2}}{2} }{ \frac{\sqrt{2}}{2} } \\[6pt] &= -1 \end{aligned} \]
Answer: \[ \sin\left(\frac{7\pi}{4}\right) = -\frac{\sqrt{2}}{2} \] \[ \cos\left(\frac{7\pi}{4}\right) = \frac{\sqrt{2}}{2} \] \[ \tan\left(\frac{7\pi}{4}\right) = -1 \]

Problem 6

Coordinates to an Angle

Identify the angle in the interval \(0\leq\theta<2\pi\), then find its exact trigonometric values.

\[ P= \left( -\frac{1}{2}, -\frac{\sqrt{3}}{2} \right) \]

Step-by-Step Solution

Both coordinates are negative, so the point lies in Quadrant III.

The coordinate magnitudes \(\dfrac{1}{2}\) and \(\dfrac{\sqrt{3}}{2}\) correspond to a \(60^\circ\), or \(\dfrac{\pi}{3}\), reference angle.

Add the reference angle to \(\pi\) to find the Quadrant III angle.

\[ \begin{aligned} \theta &= \pi+\frac{\pi}{3} \\[6pt] &= \frac{4\pi}{3} \end{aligned} \]

Read cosine and sine directly from the coordinates.

\[ \cos\theta=-\frac{1}{2} \] \[ \sin\theta= -\frac{\sqrt{3}}{2} \] \[ \tan\theta= \frac{ -\frac{\sqrt{3}}{2} }{ -\frac{1}{2} } = \sqrt{3} \]
Answer: \[ \theta=\frac{4\pi}{3} \] \[ \sin\theta= -\frac{\sqrt{3}}{2}, \qquad \cos\theta= -\frac{1}{2}, \qquad \tan\theta=\sqrt{3} \]

Solution Graph

Level 3

Reference Angles and Quadrant Signs

Use reference angles and quadrant signs to evaluate trigonometric expressions.

Problem 7

Reference Angles

Identify the quadrant and find the reference angle.

\[ \theta=\frac{11\pi}{6} \]

Step-by-Step Solution

The angle \(\dfrac{11\pi}{6}\) is between \(\dfrac{3\pi}{2}\) and \(2\pi\), so it lies in Quadrant IV.

For a Quadrant IV angle, subtract the angle from \(2\pi\).

\[ \begin{aligned} \theta_{\text{ref}} &= 2\pi-\frac{11\pi}{6} \\[6pt] &= \frac{12\pi}{6} - \frac{11\pi}{6} \\[6pt] &= \frac{\pi}{6} \end{aligned} \]
Answer: Quadrant IV with reference angle \[ \frac{\pi}{6} \]

Problem 8

Mixed Exact Values

Find each exact value.

\[ \text{a. }\cos(225^\circ) \] \[ \text{b. }\sin(120^\circ) \] \[ \text{c. }\tan(150^\circ) \]

Step-by-Step Solution

Part a: \(\cos(225^\circ)\)

The angle \(225^\circ\) lies in Quadrant III and has a \(45^\circ\) reference angle. Cosine is negative in Quadrant III.

\[ \cos(225^\circ) = -\frac{\sqrt{2}}{2} \]

Part b: \(\sin(120^\circ)\)

The angle \(120^\circ\) lies in Quadrant II and has a \(60^\circ\) reference angle. Sine is positive in Quadrant II.

\[ \sin(120^\circ) = \frac{\sqrt{3}}{2} \]

Part c: \(\tan(150^\circ)\)

The angle \(150^\circ\) lies in Quadrant II and has a \(30^\circ\) reference angle. Tangent is negative in Quadrant II.

\[ \tan(150^\circ) = -\frac{\sqrt{3}}{3} \]
Answers: \[ \text{a. } -\frac{\sqrt{2}}{2} \qquad \text{b. } \frac{\sqrt{3}}{2} \qquad \text{c. } -\frac{\sqrt{3}}{3} \]

Problem 9

Undefined Values

Find the exact value or state that the expression is undefined.

\[ \tan\left(\frac{3\pi}{2}\right) \]

Step-by-Step Solution

At \(\dfrac{3\pi}{2}\), the unit-circle point is \((0,-1)\).

Therefore,

\[ \cos\left(\frac{3\pi}{2}\right) = 0 \] \[ \sin\left(\frac{3\pi}{2}\right) = -1 \]

Tangent is sine divided by cosine.

\[ \begin{aligned} \tan\left(\frac{3\pi}{2}\right) &= \frac{ \sin\left( \frac{3\pi}{2} \right) }{ \cos\left( \frac{3\pi}{2} \right) } \\[8pt] &= \frac{-1}{0} \end{aligned} \]

Division by zero is undefined.

Answer: \[ \tan\left(\frac{3\pi}{2}\right) \text{ is undefined.} \]

Before You Check the Reference

Use the Same Process Every Time

  1. Identify the angle’s quadrant.
  2. Find its positive acute reference angle.
  3. Use the matching first-quadrant coordinate magnitudes.
  4. Apply the correct signs for the original quadrant.
  5. Read cosine and sine from the coordinates, then calculate tangent when needed.

Continue Learning

Related Unit Circle Resources

Review the lesson, keep the printable reference nearby, or continue building the triangle and angle skills used throughout trigonometry.

Unit Circle Guide

Review degrees, radians, coordinates, reference angles, and exact trigonometric values.

Unit Circle Reference PDF

Download the complete Unit Circle, exact-value table, and quadrant sign reminder.

Radians and Degrees

Learn how to convert angle measures and recognize common degree-radian equivalents.

Special Right Triangles

Review the triangle ratios that produce the Unit Circle’s exact coordinate values.

Trigonometric Graphing Lab

Explore sine, cosine, and tangent functions on an interactive coordinate plane.

Trigonometry Resources

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