Trig Values of Special Angles: Exact Table

The trig values of special angles are the exact values of the six trig functions at 30°, 45° and 60°, or $\frac{\pi}{6}$ , $\frac{\pi}{4}$ and $\frac{\pi}{3}$ in radians. They come from two special right triangles. For example, $\sin 30^\circ = \frac{1}{2}$ , $\cos 45^\circ = \frac{\sqrt{2}}{2}$ and $\tan 60^\circ = \sqrt{3}$ .

Before this lesson: You should know Six Trigonometric Functions, Special Right Triangles and Radians.

Where the Exact Values Come From

A calculator gives $\sin 30^\circ = 0.5$ but $\sin 60^\circ \approx 0.8660$ , a rounded decimal. Exact values keep the square roots, so they never lose accuracy. They come from the two triangles in special right triangles:

45°-45°-90° triangle: legs of 1 and 1, hypotenuse $\sqrt{2}$ .
30°-60°-90° triangle: short leg 1 (across from 30°), long leg $\sqrt{3}$ (across from 60°), hypotenuse 2.

Two right triangles. The left one is a 45-45-90 triangle with both legs labeled 1 and the hypotenuse labeled square root of 2. The right one is a 30-60-90 triangle with the short leg 1 across from the 30 degree angle, the long leg square root of 3 across from the 60 degree angle and the hypotenuse 2. — Read any special-angle value from these triangles. For example, sin 30° is the side across from 30° over the hypotenuse: 1/2.

Table of Special Angle Values

Angle	$\sin$	$\cos$	$\tan$	$\csc$	$\sec$	$\cot$
$30^\circ = \frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{3}}{3}$	$2$	$\frac{2\sqrt{3}}{3}$	$\sqrt{3}$
$45^\circ = \frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	$1$	$\sqrt{2}$	$\sqrt{2}$	$1$
$60^\circ = \frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2\sqrt{3}}{3}$	$2$	$\frac{\sqrt{3}}{3}$

A memory pattern: the sines of 30°, 45° and 60° are $\frac{\sqrt{1}}{2}$ , $\frac{\sqrt{2}}{2}$ and $\frac{\sqrt{3}}{2}$ . The cosines are the same list in reverse order, because 30° and 60° are complementary.

Pick the triangle: 45°-45°-90° for 45°, or 30°-60°-90° for 30° and 60°.
Label the sides opposite and adjacent to your angle.
Write the ratio for the function you need.
Rationalize any square root in the denominator.

Worked Examples

Example 1: Sine, cosine and tangent of 60°

Find the exact values of $\sin 60^\circ$ , $\cos 60^\circ$ and $\tan 60^\circ$ .

Use the 30°-60°-90° triangle. From the 60° angle, opposite = $\sqrt{3}$ , adjacent = 1, hypotenuse = 2.

\sin 60^\circ = \frac{\sqrt{3}}{2}

Sine is opposite over hypotenuse.

\cos 60^\circ = \frac{1}{2}

Cosine is adjacent over hypotenuse.

\tan 60^\circ = \frac{\sqrt{3}}{1} = \sqrt{3}

Tangent is opposite over adjacent.

Answer: $\sin 60^\circ = \frac{\sqrt{3}}{2}$ , $\cos 60^\circ = \frac{1}{2}$ , $\tan 60^\circ = \sqrt{3}$

Example 2: All six functions of π/4

Find all six trig functions of $\frac{\pi}{4}$ .

Convert to degrees: $\frac{\pi}{4} = 45^\circ$ , so use the 45°-45°-90° triangle with opposite = 1, adjacent = 1, hypotenuse = $\sqrt{2}$ .

\sin \frac{\pi}{4} = \frac{1}{\sqrt{2}}

Write sine.

\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}

Rationalize.

\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}

Cosine uses the same sides.

\tan \frac{\pi}{4} = \frac{1}{1} = 1

Write tangent.

\csc \frac{\pi}{4} = \frac{\sqrt{2}}{1} = \sqrt{2}

Flip sine.

\sec \frac{\pi}{4} = \sqrt{2}

Flip cosine.

\cot \frac{\pi}{4} = 1

Flip tangent.

Answer: $\frac{\sqrt{2}}{2}$ , $\frac{\sqrt{2}}{2}$ , 1, $\sqrt{2}$ , $\sqrt{2}$ , 1

Example 3: Evaluate an expression exactly

Find the exact value of $2 \sin 30^\circ + \tan^2 60^\circ$ .

2 \cdot \frac{1}{2} + (\sqrt{3})^2

Substitute the special values.

2 \cdot \frac{1}{2} = 1

Multiply.

(\sqrt{3})^2 = 3

Square the root.

1 + 3 = 4

Add.

Answer: 4. The notation $\tan^2 60^\circ$ means $(\tan 60^\circ)^2$ .

Example 4: Reciprocal functions in radians

Find the exact values of $\sec \frac{\pi}{6}$ and $\cot \frac{\pi}{3}$ .

Convert: $\frac{\pi}{6} = 30^\circ$ and $\frac{\pi}{3} = 60^\circ$

\sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}}

Use the reciprocal of cosine.

\frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}}

Divide by multiplying by the reciprocal.

\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}

Rationalize (see rationalizing the denominator)

\cot 60^\circ = \frac{1}{\sqrt{3}}

Use adjacent over opposite for 60°.

\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}

Rationalize.

Answer: $\sec \frac{\pi}{6} = \frac{2\sqrt{3}}{3}$ and $\cot \frac{\pi}{3} = \frac{\sqrt{3}}{3}$

Example 5: Word problem with a ladder

A 12-foot ladder leans against a wall and makes a 60° angle with the ground. How high up the wall does it reach, and how far is its base from the wall? Give exact answers, then decimals.

\sin 60^\circ = \frac{h}{12}

The ladder is the hypotenuse. The height is opposite the 60° angle, so use sine.

h = 12 \sin 60^\circ

Multiply both sides by 12.

h = 12 \cdot \frac{\sqrt{3}}{2}

Substitute the exact value.

h = 6\sqrt{3} \approx 10.39

Simplify.

\cos 60^\circ = \frac{d}{12}

The base distance is adjacent, so use cosine.

d = 12 \cdot \frac{1}{2} = 6

Multiply both sides by 12 and substitute.

Answer: The ladder reaches $6\sqrt{3} \approx 10.39$ feet up the wall, and its base is 6 feet from the wall.

Common Mistakes

Mistake	Example of the error	Fix
Swapping sine and cosine of 30° and 60°	$\sin 30^\circ = \frac{\sqrt{3}}{2}$	The side across from 30° is the short leg, 1: $\sin 30^\circ = \frac{1}{2}$ .
Putting the long leg across from 30°	Labeling the side opposite 30° as $\sqrt{3}$	The shortest side is always across from the smallest angle.
Leaving a root in the denominator	$\tan 30^\circ = \frac{1}{\sqrt{3}}$	Rationalize: $\frac{\sqrt{3}}{3}$ .
Reading tan² as tan of 2θ	$\tan^2 60^\circ = \tan 120^\circ$	$\tan^2 60^\circ$ means $(\tan 60^\circ)^2 = 3$ .
Using a rounded decimal when an exact value is asked	$\sin 60^\circ = 0.87$	Give $\frac{\sqrt{3}}{2}$ unless the problem asks for a decimal.

Related Methods

The hand trick for sine

Count up the square roots: $\sin 0^\circ = \frac{\sqrt{0}}{2}$ , $\sin 30^\circ = \frac{\sqrt{1}}{2}$ , $\sin 45^\circ = \frac{\sqrt{2}}{2}$ , $\sin 60^\circ = \frac{\sqrt{3}}{2}$ , $\sin 90^\circ = \frac{\sqrt{4}}{2}$ . The cosines run the same list backward.

Finding the angle from a value

The table also works in reverse. If $\theta$ is acute and $\cos \theta = \frac{\sqrt{2}}{2}$ , then $\theta = 45^\circ$ . This is the idea behind inverse trig functions later in the course.

Angles beyond the first quadrant

Special values also give exact values for angles like 150°, 225° and 300°. You will use reference angles and the unit circle to find those, with the signs set by the quadrant.

Practice Problems

Find the exact value of $\cos 30^\circ$ .
Find the exact value of $\tan 45^\circ$ .
Find the exact value of $\csc 60^\circ$ .
Find the exact value of $\sec 45^\circ$ .
Find the exact value of $\cot \frac{\pi}{6}$ .
Find the exact value of $\sin \frac{\pi}{4} \cos \frac{\pi}{4}$ .
Find the exact value of $\sin^2 60^\circ + \cos^2 60^\circ$ .
Find the acute angle $\theta$ , in degrees and radians, if $\cos \theta = \frac{1}{2}$ .
Find the acute angle $\theta$ , in degrees and radians, if $\tan \theta = \frac{\sqrt{3}}{3}$ .
A kite string is 40 meters long and makes a 30° angle with the ground. How high is the kite?

Show answers

From the 30° angle, adjacent = $\sqrt{3}$ and hypotenuse = 2
Answer: $\cos 30^\circ = \frac{\sqrt{3}}{2}$

\tan 45^\circ = \frac{1}{1}

In the 45°-45°-90° triangle the legs are equal.

Answer: 1

\csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{2}{\sqrt{3}}

Flip sine.

\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}

Rationalize.

Answer: $\frac{2\sqrt{3}}{3}$

\sec 45^\circ = \frac{\sqrt{2}}{1}

Flip cosine.

Answer: $\sqrt{2}$

\frac{\pi}{6} = 30^\circ

Convert.

\cot 30^\circ = \frac{\sqrt{3}}{1}

Adjacent over opposite.

Answer: $\sqrt{3}$

\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2}}{2}

Substitute.

\sqrt{2} \cdot \sqrt{2} = 2

Multiply the tops (see multiplying radicals)

2 \cdot 2 = 4

Multiply the bottoms.

\frac{2}{4} = \frac{1}{2}

Simplify.

Answer: $\frac{1}{2}$

(\frac{\sqrt{3}}{2})^2 + (\frac{1}{2})^2

Substitute.

\frac{3}{4} + \frac{1}{4}

Square each.

\frac{4}{4} = 1

Add.

Answer: 1

In the table, cosine is $\frac{1}{2}$ at 60°

60^\circ = \frac{\pi}{3}

Convert.

Answer: $\theta = 60^\circ = \frac{\pi}{3}$

In the table, tangent is $\frac{\sqrt{3}}{3}$ at 30°

30^\circ = \frac{\pi}{6}

Convert.

Answer: $\theta = 30^\circ = \frac{\pi}{6}$

10.

\sin 30^\circ = \frac{h}{40}

The height is opposite the 30° angle.

h = 40 \sin 30^\circ

Multiply both sides by 40.

h = 40 \cdot \frac{1}{2} = 20

Substitute.

Answer: 20 meters

FAQ

What are the special angles in trigonometry?

They are 30°, 45° and 60°, or $\frac{\pi}{6}$ , $\frac{\pi}{4}$ and $\frac{\pi}{3}$ . Their trig values can be written exactly with square roots.

What is the exact value of sin 60°?

$\sin 60^\circ = \frac{\sqrt{3}}{2}$ , which is about 0.866. It comes from the 30°-60°-90° triangle: the side across from 60° is $\sqrt{3}$ and the hypotenuse is 2.

How do you remember the special angle values?

Remember the two triangles, or the pattern $\frac{\sqrt{1}}{2}$ , $\frac{\sqrt{2}}{2}$ , $\frac{\sqrt{3}}{2}$ for the sines of 30°, 45° and 60°. The cosines are the same values in reverse.

Why is tan 45° equal to 1?

In a 45°-45°-90° triangle the two legs are equal. Tangent is opposite over adjacent, so it is a number divided by itself, which is 1.

Summary

The special angles are 30°, 45° and 60° ( $\frac{\pi}{6}$ , $\frac{\pi}{4}$ , $\frac{\pi}{3}$ ).
Their values come from the 45°-45°-90° and 30°-60°-90° triangles.
$\sin 30^\circ = \cos 60^\circ = \frac{1}{2}$ , $\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}$ , $\sin 60^\circ = \cos 30^\circ = \frac{\sqrt{3}}{2}$ .
$\tan 30^\circ = \frac{\sqrt{3}}{3}$ , $\tan 45^\circ = 1$ , $\tan 60^\circ = \sqrt{3}$ .
Rationalize denominators in exact answers.

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