Signs of Sine, Cosine, and Tangent
🎯 The Big Idea
The sign (positive or negative) of a trigonometric function depends entirely on which quadrant the angle lies in. Understanding this is one of the most powerful tools in trigonometry!
📐 The Unit Circle Foundation
Consider a point $P(x, y)$ on the unit circle (radius $r = 1$) at angle $\theta$ from the positive $x$-axis:
- $\cos\theta = \dfrac{x}{r} = x$ → the x-coordinate
- $\sin\theta = \dfrac{y}{r} = y$ → the y-coordinate
- $\tan\theta = \dfrac{y}{x} = \dfrac{\sin\theta}{\cos\theta}$
So the signs of sine and cosine depend on the signs of $y$ and $x$ respectively!
🗺️ The Four Quadrants
| Quadrant | Angle Range | $x$ | $y$ |
|---|---|---|---|
| I | $0° < \theta < 90°$ | $+$ | $+$ |
| II | $90° < \theta < 180°$ | $-$ | $+$ |
| III | $180° < \theta < 270°$ | $-$ | $-$ |
| IV | $270° < \theta < 360°$ | $+$ | $-$ |