Step by Step
1
What the unit circle is
A circle centered at the origin with radius exactly 1. Every point on this circle, at angle θ measured from the positive x-axis, has coordinates (cos θ, sin θ).
2
Reading off the four cardinal angles
At θ=0°: (1, 0). At θ=90°: (0, 1). At θ=180°: (−1, 0). At θ=270°: (0, −1). These four points anchor the entire unit circle and are worth memorizing exactly.
3
Combining with special right triangles
The 16 most commonly tested angles on the unit circle (multiples of 30° and 45°) all have exact coordinate values derived directly from the special right triangles (30-60-90 and 45-45-90) covered earlier.
4
Why this framework matters beyond right triangles
Unlike SOH-CAH-TOA, which only works for angles between 0° and 90° in a right triangle, the unit circle definition of sine and cosine as coordinates works for ANY angle, including negative angles and angles greater than 360°.
Applied Walkthrough
1
Find cos(180°) and sin(180°) using the unit circle. At θ=180°, the point on the unit circle is directly to the left of center, at coordinates (−1, 0).
2
Since cos(θ) is the x-coordinate: cos(180°) = −1.
3
Since sin(θ) is the y-coordinate: sin(180°) = 0.
4
This illustrates why the unit circle definition extends beyond right-triangle trigonometry — there's no actual "triangle" at 180°, yet cosine and sine are still perfectly well-defined as coordinates on the circle.
Exam Application
Exams test whether you have memorized the coordinates of the key unit circle angles (multiples of 30°, 45°, and the four cardinal directions), and whether you understand why the unit circle definition extends trig functions beyond right triangles.
⚠ Common Trap
The most common trap is mixing up which coordinate corresponds to sine and which to cosine — remember cosine is the x-coordinate (horizontal) and sine is the y-coordinate (vertical), matching alphabetical order (cosine/x both come conceptually "first" as the horizontal axis).
✓ Quick Self-Check
1. On the unit circle, what does cos(θ) represent?
The x-coordinate of the point at angle θ.
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2. On the unit circle, what does sin(θ) represent?
The y-coordinate of the point at angle θ.
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3. What are the coordinates on the unit circle at θ=180°?
(−1, 0).
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4. What are cos(180°) and sin(180°)?
cos(180°) = −1, sin(180°) = 0.
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5. Why does the unit circle definition of sine and cosine work for angles beyond 0°–90°, unlike SOH-CAH-TOA?
Because it defines sine and cosine as coordinates on a circle, which works for any angle, including negative angles and angles greater than 360°, not just angles within a right triangle.
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