Trig Functions on the Coordinate Plane
Extend trigonometry beyond right triangles by defining sin and cos for any angle using a point on the coordinate plane — including angles in all four quadrants.
A right triangle can only contain angles between 0° and 90°. But real-world applications — rotating machinery, pendulums, sound waves, satellite orbits — involve angles of any size. By placing trig functions in the coordinate plane, we unlock angles like 135°, 210°, −45°, or even 720°. This is the foundation for understanding why sine and cosine produce repeating waves, and it is what makes circular motion mathematically tractable.
- Define sin θ and cos θ using a point P(x, y) on any circle (not just the unit circle)
- Evaluate all six trig functions from a coordinate point
- Understand coterminal angles and what it means for trig functions to be periodic
- Use reference angles to find trig values in any quadrant
If the terminal side of angle θ passes through point P(x, y) at distance \(r = \sqrt{x^2 + y^2}\) from the origin:
\[ \sin\theta = \frac{y}{r} \qquad \cos\theta = \frac{x}{r} \qquad \tan\theta = \frac{y}{x} \]The unit circle is the special case where \(r = 1\), giving \(\sin\theta = y\) and \(\cos\theta = x\). For any other circle of radius r, divide by r to normalize.
Coterminal Angles — Same Terminal Side, Different Rotations
| Angle | Coterminal (+ 360°) | Coterminal (− 360°) |
|---|---|---|
| 45° | 405° | −315° |
| 120° | 480° | −240° |
| −30° | 330° | −390° |
All coterminal angles have identical sin, cos, and tan values.
The terminal side of θ passes through P(3, 4). Find sin θ, cos θ, and tan θ.
- \(r = \sqrt{3^2 + 4^2} = \sqrt{25} = 5\)
- \(\sin\theta = 4/5 = 0.8\), \(\cos\theta = 3/5 = 0.6\), \(\tan\theta = 4/3 \approx 1.333\)
P(−5, −12) is on the terminal side of θ. Find all six trig values.
- \(r = \sqrt{25 + 144} = \sqrt{169} = 13\)
- \(\sin\theta = -12/13\), \(\cos\theta = -5/13\), \(\tan\theta = (-12)/(-5) = 12/5\)
- Reciprocals: \(\csc\theta = -13/12\), \(\sec\theta = -13/5\), \(\cot\theta = 5/12\)
A motor shaft rotates 930° from its starting position. What equivalent angle between 0° and 360° describes the same shaft position? What are the sin and cos at that position?
- \(930° - 2(360°) = 930° - 720° = 210°\) — the shaft is at 210°.
- Reference angle = 210° − 180° = 30° (QIII).
- \(\sin210° = -\sin30° = -0.5\) and \(\cos210° = -\cos30° = -\frac{\sqrt{3}}{2}\).
- Terminal side passes through P(−3, 4). Find r and evaluate sin θ and cos θ.
- Find the coterminal angle between 0° and 360° for −100°.
- In which quadrant is the angle 500°? What is its reference angle?
▶ Show Answers
- \(r = \sqrt{9+16} = 5\); \(\sin\theta = 4/5\), \(\cos\theta = -3/5\) (QII).
- \(-100° + 360° = \mathbf{260°}\).
- \(500° - 360° = 140°\) → Quadrant II; ref angle = \(180° - 140° = \mathbf{40°}\).
- Forgetting that r is always positive: r = √(x² + y²) is always positive. The signs of sin θ and cos θ come from the signs of y and x, not from r.
- Not reducing to the standard range first: For 750°, find the coterminal angle first (750° − 2×360° = 30°), then evaluate sin(30°) rather than trying to reason about 750° directly.
- Confusing period with coterminal: Coterminal angles add full rotations (360°). Periodicity is the mathematical property that says trig functions repeat — these are the same idea but expressed differently.
- For any point P(x, y) at distance r: \(\sin\theta = y/r\), \(\cos\theta = x/r\), \(\tan\theta = y/x\).
- Coterminal angles are equal modulo 360°. Trig functions are periodic with period 360° (sin/cos) or 180° (tan).
- Use reference angles to find trig values; use ASTC to determine the sign.
Rotating machinery (turbines, motors, crankshafts) is analyzed using angles that exceed 360° — because the shaft has completed multiple full rotations. Engineers use modular arithmetic (coterminal angles) to reduce these to the standard range, then apply sin and cos to find torque, displacement, and velocity at any point in the rotation cycle. This is the math of every engine, generator, and pump in the world.
Calculator Connection
The Trigonometric Functions Calculator evaluates all six trig functions for any angle in any quadrant. The Coterminal & Reference Angle Calculator reduces any angle to its standard position.
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Trig Functions on the Coordinate Plane — Quiz
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