Trig Functions on the Coordinate Plane
Calculate sine, cosine, and tangent for any angle in all four quadrants by using x, y, and r.
Beyond right triangles, trigonometry works for any point \((x, y)\) on a plane. The distance from the origin is \(r = \sqrt{x^2 + y^2}\).
\[ \sin(\theta) = \frac{y}{r} \qquad \cos(\theta) = \frac{x}{r} \qquad \tan(\theta) = \frac{y}{x} \]The Four Quadrants
Depending on where the point is, the ratios may be positive or negative. Use the acronym ASTC (All Students Take Calculus):
- Quadrant I: All ratios are positive (+, +)
- Quadrant II: Sine is positive (-, +)
- Quadrant III: Tangent is positive (-, -)
- Quadrant IV: Cosine is positive (+, -)
Find \(\sin(\theta)\) for the point \((-3, 4)\).
\[ r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = 5 \] \[ \sin(\theta) = \frac{y}{r} = \frac{4}{5} = 0.8 \]Find \(\cos(\theta)\) for the point \((-5, -12)\).
\[ r = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = 13 \] \[ \cos(\theta) = \frac{x}{r} = \frac{-5}{13} \approx -0.385 \]When working in different quadrants, always look for the reference angle between the terminal side and the nearest x-axis. This angle is always positive and between 0 and 90°.
A robot arm knows its position on a plane using (x, y) coordinates. To rotate to a specific object, the robot's brain uses Arctan with the current coordinates to find the correct motor angle in the right quadrant.
Calculator Connection
The Trig Functions calculator allows you to input any angle from 0° to 360° (or beyond) and instantly find the correct positive or negative ratio.
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