Trig Values of Special Angles

Trig Values of Special Angles
Exact Values of Trig Functions

Certain angles have trig values that may be computed exactly. Of these, the angles listed below are some of the angles most commonly used in math classes.

$Table of trig values for 0°,30°,45°,60°,90°: sin, cos, tan, csc, sec, cot with exact values using fractions and √2,√3.$

Key Formula

\begin{array}{c|ccc} \theta & \sin\theta & \cos\theta & \tan\theta \\ \hline 0° \;(0) & 0 & 1 & 0 \\ 30° \;\left(\tfrac{\pi}{6}\right) & \tfrac{1}{2} & \tfrac{\sqrt{3}}{2} & \tfrac{\sqrt{3}}{3} \\ 45° \;\left(\tfrac{\pi}{4}\right) & \tfrac{\sqrt{2}}{2} & \tfrac{\sqrt{2}}{2} & 1 \\ 60° \;\left(\tfrac{\pi}{3}\right) & \tfrac{\sqrt{3}}{2} & \tfrac{1}{2} & \sqrt{3} \\ 90° \;\left(\tfrac{\pi}{2}\right) & 1 & 0 & \text{undefined} \end{array}

Where:

$\theta$ = The angle, measured in degrees or radians
$\sin\theta$ = The sine of the angle (opposite over hypotenuse in a right triangle)
$\cos\theta$ = The cosine of the angle (adjacent over hypotenuse in a right triangle)
$\tan\theta$ = The tangent of the angle (sine divided by cosine)

Worked Example

Problem: Find the exact value of sin 60° + cos 30°.

Step 1: Look up sin 60° from the special angles table.

\sin 60° = \frac{\sqrt{3}}{2}

Step 2: Look up cos 30° from the special angles table.

\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 ≈ 1.732

Another Example

This example extends beyond first-quadrant angles by using a reference angle and the sign convention from ASTC (All Students Take Calculus) to evaluate a trig function in Quadrant II.

Problem: Evaluate tan(5π/6) using special angle values and reference angles.

Step 1: Determine the quadrant. 5π/6 is in Quadrant II (between π/2 and π).

\frac{5\pi}{6} = \pi - \frac{\pi}{6}

Step 2: Find the reference angle. Subtract the angle from π.

\text{Reference angle} = \pi - \frac{5\pi}{6} = \frac{\pi}{6}

Step 3: Recall the tangent of the reference angle π/6 (which is 30°).

\tan\frac{\pi}{6} = \frac{\sqrt{3}}{3}

Step 4: Apply the sign rule. Tangent is negative in Quadrant II (since sine is positive and cosine is negative there).

\tan\frac{5\pi}{6} = -\frac{\sqrt{3}}{3}

Answer: tan(5π/6) = −√3/3

Frequently Asked Questions

How do you memorize the trig values of special angles?

A popular trick uses the pattern √0/2, √1/2, √2/2, √3/2, √4/2 for sin 0°, sin 30°, sin 45°, sin 60°, and sin 90°. This simplifies to 0, 1/2, √2/2, √3/2, and 1. Cosine follows the same values in reverse order. You can also derive them from the 30-60-90 and 45-45-90 triangles.

Why is tan 90° undefined?

Tangent equals sine divided by cosine. At 90°, sin 90° = 1 and cos 90° = 0, so you would be dividing by zero. Division by zero is undefined, which is why tan 90° has no finite value.

What are the special angle trig values for the reciprocal functions (csc, sec, cot)?

The reciprocal functions are csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ. For example, csc 30° = 1/(1/2) = 2, sec 60° = 1/(1/2) = 2, and cot 45° = 1/1 = 1. Wherever the original function is 0, the reciprocal is undefined.

Exact trig values (special angles) vs. Calculator approximations

	Exact trig values (special angles)	Calculator approximations
Form of answer	Exact expressions with radicals, e.g. √3/2	Decimal approximations, e.g. 0.8660...
Which angles	Only specific angles (0°, 30°, 45°, 60°, 90° and their multiples)	Any angle at all
Precision	Perfectly exact — no rounding	Rounded to a finite number of decimal places
When to use	Algebra, proofs, and exams requiring exact answers	Applied problems where a numerical result is needed

Why It Matters

These values appear constantly in precalculus, calculus, physics, and engineering courses. You need them to simplify expressions, solve trig equations, and evaluate limits and integrals without a calculator. Many standardized tests and college exams explicitly require exact values, so memorizing or quickly deriving them saves significant time.

Common Mistakes

Mistake: Swapping sin and cos values — for instance, writing sin 30° = √3/2 instead of 1/2.

Correction: Remember that sine starts small and grows: sin 0° = 0, sin 30° = 1/2, sin 60° = √3/2. Cosine does the opposite: cos 0° = 1, cos 30° = √3/2, cos 60° = 1/2. The two functions mirror each other across the table.

Mistake: Forgetting to apply the correct sign when the angle is outside the first quadrant.

Correction: Use the ASTC rule: All trig functions are positive in Quadrant I, only Sine in II, only Tangent in III, only Cosine in IV. Always find the reference angle first, then attach the appropriate sign.

Related Terms

Special Angles — The angles whose trig values are memorized
Sine — One of the primary trig functions evaluated
Cosine — One of the primary trig functions evaluated
Tangent — Sine divided by cosine for each angle
Trig Functions — The family of functions these values belong to
Radian — Alternate unit for measuring these angles
Degree — Common unit for measuring these angles
Angle — The geometric quantity being measured