Special Angles — Trig Values, Table & Examples

Special Angles

Some of the angles for which the exact values of trig functions are known are listed below. These special angles from 0° to 360° are listed below with their radian equivalents.

Two-column table converting degrees to radians for angles 0°–360°, including 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°,...

See also

Trigonometry

Key Formula

\begin{array}{c|c|c|c|c} \theta & 0° & 30° & 45° & 60° & 90° \\ \hline \text{Radians} & 0 & \dfrac{\pi}{6} & \dfrac{\pi}{4} & \dfrac{\pi}{3} & \dfrac{\pi}{2} \\ \hline \sin\theta & 0 & \dfrac{1}{2} & \dfrac{\sqrt{2}}{2} & \dfrac{\sqrt{3}}{2} & 1 \\ \hline \cos\theta & 1 & \dfrac{\sqrt{3}}{2} & \dfrac{\sqrt{2}}{2} & \dfrac{1}{2} & 0 \\ \hline \tan\theta & 0 & \dfrac{\sqrt{3}}{3} & 1 & \sqrt{3} & \text{undef.} \end{array}

Where:

$\theta$ = The special angle, measured in degrees or radians
$\sin\theta$ = The sine of the angle (opposite over hypotenuse)
$\cos\theta$ = The cosine of the angle (adjacent over hypotenuse)
$\tan\theta$ = The tangent of the angle (opposite over adjacent, or sin/cos)

Worked Example

Problem: Find the exact value of sin(60°) + cos(30°) without using a calculator.

Step 1: Recall the exact sine value for the special angle 60°.

\sin 60° = \frac{\sqrt{3}}{2}

Step 2: Recall the exact cosine value for the special angle 30°.

\cos 30° = \frac{\sqrt{3}}{2}

Step 3: Add the two values together.

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

Answer: sin(60°) + cos(30°) = √3

Another Example

This example extends special angles beyond the first quadrant (0°–90°) by using a reference angle and quadrant sign rules, which is how special angles are most commonly applied in practice.

Problem: Find the exact value of tan(225°).

Step 1: Determine the reference angle. 225° is in the third quadrant (between 180° and 270°). Subtract 180° to find the reference angle.

225° - 180° = 45°

Step 2: Recall that tan(45°) = 1 from the special angles table.

\tan 45° = 1

Step 3: Determine the sign. In the third quadrant, both sine and cosine are negative, so tangent (sin/cos) is positive.

\tan 225° = +\tan 45° = 1

Answer: tan(225°) = 1

Frequently Asked Questions

What are the special angles in trigonometry?

The special angles are 0°, 30°, 45°, 60°, and 90° (equivalently 0, π/6, π/4, π/3, and π/2 radians), along with their corresponding angles in the second, third, and fourth quadrants (such as 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, and 360°). These are called "special" because their trigonometric values can be expressed as exact fractions involving √2 and √3 rather than infinite decimals.

How do you memorize the special angle values?

A popular method is the "counting pattern." For sine, write 0°, 30°, 45°, 60°, 90° and assign the numerators 0, 1, 2, 3, 4 under a square root, all over 2: √0/2, √1/2, √2/2, √3/2, √4/2. This gives 0, 1/2, √2/2, √3/2, 1. For cosine, the same values appear in reverse order. You can also derive the 30° and 60° values from a 30-60-90 triangle with sides 1, √3, and 2, and the 45° values from a 45-45-90 triangle with sides 1, 1, and √2.

Why are special angles important?

Special angles let you compute exact trigonometric values without a calculator, which is essential for simplifying expressions in algebra, calculus, and physics. Many standardized tests and college courses require you to know these values from memory. They also form the basis for understanding the unit circle and solving trigonometric equations analytically.

Special Angles (Degrees) vs. Special Angles (Radians)

	Special Angles (Degrees)	Special Angles (Radians)
0°	0°	0
30°	30°	π/6
45°	45°	π/4
60°	60°	π/3
90°	90°	π/2
180°	180°	π
270°	270°	3π/2
360°	360°	2π
When to use	Geometry problems and introductory trig courses	Calculus, advanced math, and any formula requiring radian measure

Why It Matters

Special angles appear constantly in geometry proofs, physics problems involving forces and vectors, and throughout calculus when you evaluate integrals and derivatives of trigonometric functions. On standardized tests like the SAT, ACT, and AP exams, knowing these exact values saves significant time and prevents rounding errors. Mastering special angles also gives you a deep understanding of the unit circle, which is the single most important tool in trigonometry.

Common Mistakes

Mistake: Swapping sine and cosine values for 30° and 60°. For example, writing sin(30°) = √3/2 instead of 1/2.

Correction: Remember that sin(30°) = 1/2 and cos(30°) = √3/2. A helpful check: sine starts small (0 at 0°) and grows to 1 at 90°, so sin(30°) should be smaller than sin(60°). Since 1/2 < √3/2, the values are sin(30°) = 1/2 and sin(60°) = √3/2.

Mistake: Forgetting to apply the correct sign when using special angles outside the first quadrant.

Correction: Use the mnemonic "All Students Take Calculus" (ASTC) to remember which trig functions are positive in each quadrant: All in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4. First find the reference angle, then apply the appropriate sign.

Related Terms

Angle — The fundamental geometric concept behind special angles
Trig Values of Special Angles — Complete table of exact trig values
Radian — Alternative unit for measuring special angles
Trigonometry — The broader field where special angles are used
Unit Circle — Visual model placing all special angles on a circle
Reference Angle — Used to extend special angles beyond the first quadrant
Sine — Key trig function evaluated at special angles
Cosine — Key trig function evaluated at special angles