Trigonometry Unit 3 — Unit Circle.
This unit covers unit circle values, reference angles and trig values of any angle — essential concepts for Trigonometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers unit circle values, reference angles and trig values of any angle — essential concepts for Trigonometry. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Unit Circle Values
The unit circle is a circle with radius 1 centered at the origin. Every point on the unit circle has coordinates (cos θ, sin θ), where θ is the angle measured counterclockwise from the positive x-axis. Students must memorize the exact coordinates at the 16 standard angles (multiples of 30° and 45°).
Key Points
- At angle θ, the x-coordinate = cos θ and the y-coordinate = sin θ
- Key angles in degrees and radians: 0°=0, 30°=π/6, 45°=π/4, 60°=π/3, 90°=π/2, and their continuations through 360°
- sin is positive in Q1 and Q2; cos is positive in Q1 and Q4 (use ASTC: All Students Take Calculus)
- tan θ = sin θ / cos θ; tan is undefined when cos θ = 0 (at 90° and 270°)
Find the exact value of sin(5π/6).
5π/6 is in Quadrant II, and its reference angle is π − 5π/6 = π/6. Since sin is positive in Q II, sin(5π/6) = sin(π/6) = 1/2. No calculator needed — just identify the quadrant and reference angle.
2 Reference Angles
A reference angle is the acute angle (between 0° and 90°) formed between the terminal side of an angle and the x-axis. Reference angles allow you to reduce any angle to a first-quadrant equivalent so you can apply known unit circle values.
Key Points
- Reference angle is always positive and always between 0° and 90° (or 0 and π/2)
- Q I: ref angle = θ | Q II: ref angle = 180° − θ | Q III: ref angle = θ − 180° | Q IV: ref angle = 360° − θ
- For negative angles or angles > 360°, first find the coterminal angle between 0° and 360°, then find the reference angle
- The trig value of any angle equals ± the trig value of its reference angle; the sign depends on the quadrant
Find the reference angle for 240°.
240° is in Quadrant III (between 180° and 270°). Using the Q III formula: reference angle = 240° − 180° = 60°. So any trig function of 240° will equal ± the same trig function of 60°, with the sign determined by which quadrant 240° is in.
3 Trig Values Of Any Angle
To find the exact trig value of any angle, find its reference angle, look up the unit circle value for that reference angle, then assign the correct sign based on which quadrant the original angle is in. This process works for degrees, radians, negative angles, and angles beyond 360°.
Key Points
- Step 1: Find the coterminal angle in [0°, 360°) if the angle is negative or greater than 360°
- Step 2: Identify the quadrant and find the reference angle
- Step 3: Write the trig value of the reference angle, then apply the correct sign (ASTC rule)
- Reciprocal functions: csc = 1/sin, sec = 1/cos, cot = 1/tan — undefined when the denominator is 0
Find the exact value of cos(−π/3).
A negative angle means clockwise rotation, so −π/3 is coterminal with 2π − π/3 = 5π/3, which is in Quadrant IV. The reference angle is 2π − 5π/3 = π/3. Cosine is positive in Q IV, and cos(π/3) = 1/2, so cos(−π/3) = 1/2.
Questions, answered.
What is Unit Circle?
Unit Circle is Unit 3 of Trigonometry, covering unit circle values, reference angles and trig values of any angle.
How to study for Trigonometry Unit 3?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.