Angles in standard position start on the positive x-axis and rotate around the origin, which makes the unit circle a powerful tool for trigonometry. A reference angle is the small positive angle between the terminal side of an angle and the x-axis. A coterminal angle is any angle that lands on the same terminal side after adding or subtracting full rotations.
These ideas help you simplify large, negative, or unfamiliar angles into angles whose trig values are easier to recognize.
Key Facts
- Coterminal angles in degrees: θ + 360k, where k is any integer.
- Coterminal angles in radians: θ + 2πk, where k is any integer.
- Reference angle in Quadrant I: α = θ.
- Reference angle in Quadrant II: α = 180° - θ, or α = π - θ.
- Reference angle in Quadrant III: α = θ - 180°, or α = θ - π.
- Reference angle in Quadrant IV: α = 360° - θ, or α = 2π - θ.
Vocabulary
- Standard position
- An angle is in standard position when its vertex is at the origin and its initial side lies on the positive x-axis.
- Terminal side
- The terminal side is the ray where an angle ends after rotating from its initial side.
- Reference angle
- A reference angle is the acute angle formed between an angle's terminal side and the x-axis.
- Coterminal angles
- Coterminal angles are angles in standard position that share the same terminal side.
- Unit circle
- The unit circle is a circle of radius 1 centered at the origin, used to connect angles with sine, cosine, and tangent values.
Common Mistakes to Avoid
- Using the angle itself as the reference angle in every quadrant is wrong because reference angles must be measured to the nearest x-axis and are usually acute.
- Forgetting to reduce a large angle before finding its quadrant is wrong because angles like 750° are easier to analyze after subtracting full rotations.
- Ignoring the sign of trig values is wrong because the reference angle gives the size of the trig value, but the quadrant determines whether it is positive or negative.
- Adding 180° instead of 360° to find coterminal angles is wrong because a full rotation is 360°, while 180° usually points in the opposite direction.
Practice Questions
- 1 Find one positive and one negative coterminal angle for 125°.
- 2 Find the reference angle for 240°, then determine the signs of sin 240°, cos 240°, and tan 240°.
- 3 Explain why 45°, 405°, and -315° have the same sine and cosine values, using the idea of coterminal angles.