Special angles are angles whose sine, cosine, and tangent values can be found exactly without a calculator. The most common ones are 0°, 30°, 45°, 60°, and 90°, along with their radian measures 0, π/6, π/4, π/3, and π/2. These values appear constantly in geometry, physics, engineering, and calculus.
Knowing them makes it easier to solve triangles, graph trig functions, and check calculator answers.
The unit circle connects each angle to a point (cos θ, sin θ), so cosine is the x-coordinate and sine is the y-coordinate. The 30-60-90 and 45-45-90 reference triangles explain why the exact values contain 1/2, √2/2, and √3/2. Tangent comes from the ratio sin θ / cos θ, as long as cos θ is not zero.
Reference angles let you extend the same special-angle values to other quadrants by choosing the correct positive or negative sign.
Key Facts
- Unit circle point: (x, y) = (cos θ, sin θ).
- Special angle radians: 0° = 0, 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2.
- sin values for 0°, 30°, 45°, 60°, 90°: 0, 1/2, √2/2, √3/2, 1.
- cos values for 0°, 30°, 45°, 60°, 90°: 1, √3/2, √2/2, 1/2, 0.
- tan θ = sin θ / cos θ, so tan values for 0°, 30°, 45°, 60° are 0, √3/3, 1, √3.
- tan 90° is undefined because tan 90° = sin 90° / cos 90° = 1 / 0.
Vocabulary
- Unit circle
- A circle with radius 1 centered at the origin, used to define sine and cosine for any angle.
- Special angle
- An angle such as 30°, 45°, or 60° whose trigonometric values can be written exactly.
- Radian
- A unit of angle measure based on arc length, where 180° equals π radians.
- Reference angle
- The acute angle formed between the terminal side of an angle and the x-axis.
- Tangent
- A trigonometric ratio defined by tan θ = sin θ / cos θ when cos θ is not zero.
Common Mistakes to Avoid
- Swapping sine and cosine values for 30° and 60° is wrong because sin 30° = 1/2 while cos 30° = √3/2, and the values reverse at 60°.
- Writing tan 90° = 0 is wrong because tan 90° requires division by cos 90°, and cos 90° = 0.
- Forgetting to convert degrees to radians is wrong when a problem asks for exact radian measures, since 60° should be written as π/3, not 60.
- Ignoring quadrant signs is wrong because reference angles give the size of the trig value, but the quadrant determines whether sine, cosine, and tangent are positive or negative.
Practice Questions
- 1 Find the exact values of sin 60°, cos 60°, and tan 60°.
- 2 Convert 30°, 45°, and 90° to radians, then write the unit circle point for each angle.
- 3 An angle in Quadrant II has a reference angle of 30°. Explain which of sin θ, cos θ, and tan θ are positive or negative, and give their exact values.