Angles in standard position start on the positive x-axis and rotate around the origin. Coterminal angles help us describe the same final direction using different amounts of rotation. Reference angles make trigonometry easier by connecting any angle to a small acute angle near the x-axis.
These ideas are important for graphing, unit circle work, periodic motion, and solving trigonometric equations.
Key Facts
- Coterminal angles differ by a full rotation: θ + 360°k, where k is any integer.
- In radians, coterminal angles differ by 2π: θ + 2πk, where k is any integer.
- A reference angle is always the positive acute angle between the terminal side and the x-axis.
- Quadrant I reference angle: α = θ for 0° < θ < 90°.
- Quadrant II reference angle: α = 180° - θ; Quadrant III: α = θ - 180°; Quadrant IV: α = 360° - θ.
- Trig functions use the reference angle for size and the quadrant for sign, such as sin 210° = -sin 30°.
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 showing where the angle ends after rotating from the initial side.
- Coterminal angles
- Coterminal angles are angles in standard position that share the same terminal side.
- Reference angle
- A reference angle is the positive acute angle formed between an angle's terminal side and the x-axis.
- Quadrant
- A quadrant is one of the four regions of the coordinate plane formed by the x-axis and y-axis.
Common Mistakes to Avoid
- Forgetting that negative angles rotate clockwise is wrong because angle direction changes where the terminal side lands before finding a coterminal angle.
- Using the y-axis to find the reference angle is wrong because a reference angle is always measured to the x-axis.
- Leaving a reference angle as obtuse is wrong because reference angles must be positive and acute, except for special axis angles where the reference angle can be 0° or 90°.
- Ignoring the quadrant sign is wrong because the reference angle gives the size of a trig value, but the quadrant determines whether sine, cosine, or tangent is positive or negative.
Practice Questions
- 1 Find two positive and two negative coterminal angles for 75°.
- 2 Find the reference angle for 250° and determine the signs of sin 250°, cos 250°, and tan 250°.
- 3 An angle of -120° and an angle of 240° have the same terminal side. Explain why they are coterminal and describe how their reference angle is used in trigonometry.