Trig Values of Special Angles

Certain angles come up so frequently in trigonometry that you are expected to know their exact sine, cosine, and tangent values without a calculator. These are the special angles: 0°, 30°, 45°, 60°, and 90°. Their exact values come from two special right triangles.

The Two Special Triangles

The 45-45-90 Triangle

Start with a square with side length 1. Cut it diagonally. You get two right triangles, each with:

• Two legs of length 1
• A hypotenuse of length $\sqrt{2}$ (by the Pythagorean theorem: $\sqrt{1^2 + 1^2} = \sqrt{2}$)
• Two 45-degree angles and one 90-degree angle

From this triangle:

$\sin 45° = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0.707$

$\cos 45° = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0.707$

$\tan 45° = \frac{1}{1} = 1$

Notice that sine and cosine of 45° are equal. This makes sense — the triangle is isosceles.

The 30-60-90 Triangle

Start with an equilateral triangle with side length 2. Cut it in half vertically. You get two right triangles, each with:

• A short leg of length 1 (half the base, opposite the 30° angle)
• A long leg of length $\sqrt{3}$ (the altitude, opposite the 60° angle)
• A hypotenuse of length 2 (the original side)

From this triangle, using the 30° angle:

$\sin 30° = \frac{1}{2} = 0.5 \qquad \cos 30° = \frac{\sqrt{3}}{2} \approx 0.866 \qquad \tan 30° = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \approx 0.577$

Using the 60° angle:

$\sin 60° = \frac{\sqrt{3}}{2} \approx 0.866 \qquad \cos 60° = \frac{1}{2} = 0.5 \qquad \tan 60° = \frac{\sqrt{3}}{1} = \sqrt{3} \approx 1.732$

Notice the symmetry: $\sin 30° = \cos 60°$ and $\sin 60° = \cos 30°$. This is always true for complementary angles.

The Complete Reference Table

This is the table to memorize. It appears on nearly every trig exam.

Memory trick for sine values: The sine values for 0°, 30°, 45°, 60°, 90° follow the pattern:

$\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$

which simplifies to $0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1$. The cosine values are the same list in reverse order.

Worked Examples

Example 1: Exact Calculation Without a Calculator

Find the exact value of $\sin 60° \times \cos 30°$.

From the table: $\sin 60° = \frac{\sqrt{3}}{2}$ and $\cos 30° = \frac{\sqrt{3}}{2}$.

$\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2} = \frac{3}{4}$

Answer: $\frac{3}{4}$ (exact).

Example 2: Finding a Side Using Special Angle Values

A right triangle has a 30-degree angle and a hypotenuse of 10. Find the exact lengths of both legs.

The side opposite 30° is: $O = 10 \times \sin 30° = 10 \times \frac{1}{2} = 5$

The side adjacent to 30° is: $A = 10 \times \cos 30° = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3}$

Verification: $5^2 + (5\sqrt{3})^2 = 25 + 75 = 100 = 10^2$. Correct.

Answer: The legs are exactly 5 and $5\sqrt{3} \approx 8.66$.

Example 3: Using Special Angles in the Unit Circle

What are the coordinates of the point at 150° on the unit circle?

150° is in Quadrant II. The reference angle is $180° - 150° = 30°$.

In Q II, cosine is negative and sine is positive:

$\cos 150° = -\cos 30° = -\frac{\sqrt{3}}{2}$

$\sin 150° = \sin 30° = \frac{1}{2}$

Answer: The coordinates are $\left(-\frac{\sqrt{3}}{2}, \frac{1}{2}\right)$.

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• The 45-45-90 triangle (legs 1, 1, hypotenuse $\sqrt{2}$) gives the trig values for 45°
• The 30-60-90 triangle (legs 1, $\sqrt{3}$, hypotenuse 2) gives the trig values for 30° and 60°
• Sine values for 0°-90° follow the pattern: $0, \frac{1}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, 1$; cosine is the same list reversed
• Complementary angles: $\sin\theta = \cos(90° - \theta)$
• Memorize the reference table — it is required knowledge for SAT, ACT, and any trig course

Return to Trigonometry for more topics in this section.