Welcome to Trigonometry: Unit Circle Training
Course Duration: 30–40 minutes
Trigonometry connects angles to lengths. It shows up in low-voltage work whenever you bend a conduit offset, aim a camera or antenna, find a cable run along a slope, or check a layout. The unit circle is the simplest way to understand where the trig values come from and how their signs change.
What You'll Learn:
- Describe the unit circle and read cosine and sine as the x- and y-coordinates of a point
- Convert between degrees and radians, and set your calculator to the right mode
- Recall exact values for the special angles, quadrant signs, and reference angles
- Solve right triangles with SOH–CAH–TOA, inverse trig, and the Pythagorean theorem
- Apply trig to a conduit offset bend
Course Structure
- Introduction to the Unit Circle
- Degrees and Radians
- Key Angles, Quadrants, and Reference Angles
- Solving Right Triangles
- Final Assessment — 10 questions, 80% (8 of 10) to pass
Each section ends with a knowledge check: you get two attempts, and a first miss gives you a hint rather than the answer. You finish the check to unlock the next section. You'll enter your name and email just before you submit the final assessment. Your results are then emailed automatically to the training administrator, with a copy to you, whether or not you pass.
Section 1: Introduction to the Unit Circle
The unit circle is a circle centered at the origin (0, 0) with a radius of exactly 1. Its equation is x² + y² = 1. Because the radius is 1, the coordinates of any point on the circle are the cosine and sine of its angle.
Angles in standard position
An angle in standard position starts on the positive x-axis. Counterclockwise rotation is positive; clockwise is negative. The point where the angle's terminal side crosses the circle is (x, y):
Why it's useful
- Gives exact values of sine and cosine for the common angles
- Shows the sign of each function in every quadrant
- Shows that the functions repeat every 360° (they're periodic) — sin(θ + 360°) = sin θ
- Since every point satisfies x² + y² = 1, it gives the identity sin²θ + cos²θ = 1
Section 2: Degrees and Radians
Angles are measured in degrees or radians. A full circle is 360°, a convention that dates back to ancient Babylonian astronomy. One radian is the angle whose arc length equals the radius, and it's the standard unit in mathematics, physics, and most calculators' advanced functions.
| Fraction of a circle | Degrees | Radians |
|---|---|---|
| Full | 360° | 2π |
| Half | 180° | π |
| Quarter | 90° | π/2 |
| One radian | ≈ 57.3° | 1 |
Common angles
| Degrees | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| Radians | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
Section 3: Key Angles, Quadrants, and Reference Angles
Quadrant I special angles
| Angle | Point (cos, sin) | tan |
|---|---|---|
| 0° (0) | (1, 0) | 0 |
| 30° (π/6) | (√3/2, 1/2) | √3/3 (≈ 0.577) |
| 45° (π/4) | (√2/2, √2/2) | 1 |
| 60° (π/3) | (1/2, √3/2) | √3 (≈ 1.732) |
| 90° (π/2) | (0, 1) | undefined |
Signs by quadrant
| Quadrant | Angles | cos (x) | sin (y) | tan |
|---|---|---|---|---|
| I | 0°–90° | + | + | + |
| II | 90°–180° | − | + | − |
| III | 180°–270° | − | − | + |
| IV | 270°–360° | + | − | − |
A common memory aid is “All Students Take Calculus”: All are positive in QI, Sine in QII, Tangent in QIII, Cosine in QIV.
Reference angles
The reference angle is the acute angle between the terminal side and the x-axis. Find the value for the reference angle, then apply the quadrant's sign.
| Quadrant | Reference angle |
|---|---|
| II | 180° − θ |
| III | θ − 180° |
| IV | 360° − θ |
Section 4: Solving Right Triangles
In a right triangle, the sides are named relative to the angle θ you're working with. When the hypotenuse is 1, the triangle sits exactly on the unit circle, so these ratios are the same values you learned in Section 3.
Worked examples
The inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) turn a ratio back into an angle.
Final Assessment
Answer all 10 questions. You need 80% (8 of 10) to pass. These questions are different from the section knowledge checks. Enter your name and email below the questions, then submit.
Your Information
When you submit, your results are emailed automatically to the training administrator, and a copy is sent to the email address you enter here — whether or not you pass.
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