18 min read

Special Angles

The exact sin, cos, and tan values for 0°, 30°, 45°, 60°, and 90° come up again and again.

Why These Angles Matter

Five angles appear so often in trigonometry that it helps to know their exact values without a calculator: 0 °, 30 °, 45 °, 60 °, and 90 °. Two special right triangles give most of those exact values: 45 ° - 45 ° - 90 ° and 30 ° - 60 ° - 90 ° .

Special right triangles

Left: equal legs \to 45 °. Right: short leg opposite 30 °, long leg opposite 60 °, hypotenuse twice the short leg.

These are exact values. sin and cos swap between 30 ° and 60 °. tan(90 °) is — (not defined) because cos(90 °) = 0 .

Example 1 — using a special angle directly

Find the exact value of sin (3 0^{\circ}) + cos (6 0^{\circ}) . 1 sin (3 0^{\circ}) = \frac{1}{2} 2 cos (6 0^{\circ}) = \frac{1}{2} 3 Add: \frac{1}{2} + \frac{1}{2} = 1 Answer: 1

Practice

Find sin(45 °) \times cos(45 °). Round to 2 decimal places.

Your answer

Why These Angles Matter

Five angles appear so often in trigonometry that it helps to know their exact values without a calculator: 0 °, 30 °, 45 °, 60 °, and 90 °. Two special right triangles give most of those exact values: 45 ° - 45 ° - 90 ° and 30 ° - 60 ° - 90 ° .

Special right triangles

Left: equal legs \to 45 °. Right: short leg opposite 30 °, long leg opposite 60 °, hypotenuse twice the short leg.

These are exact values. sin and cos swap between 30 ° and 60 °. tan(90 °) is — (not defined) because cos(90 °) = 0 .

Example 1 — using a special angle directly

Find the exact value of sin (3 0^{\circ}) + cos (6 0^{\circ}) . 1 sin (3 0^{\circ}) = \frac{1}{2} 2 cos (6 0^{\circ}) = \frac{1}{2} 3 Add: \frac{1}{2} + \frac{1}{2} = 1 Answer: 1

Practice

Find sin(45 °) \times cos(45 °). Round to 2 decimal places.

Your answer