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Trig Ratios of Any Angle
The sign of sin, cos, and tan depends on which quadrant the angle falls in.
The Four Quadrants
The unit circle is divided into four quadrants by the horizontal and vertical axes. Moving anti-clockwise from °:
- Quadrant : ° to °
- Quadrant : ° to °
- Quadrant : ° to °
- Quadrant : ° to °
In each quadrant, the -coordinate (cos) and -coordinate (sin) can be positive or negative, which changes the sign of each ratio.
| Which ratios are positive in each quadrant ("All Students Take Calculus") | |
|---|---|
Example 1 — using the ASTC rule
Is sin(°) positive or negative?
- ° falls in Quadrant (between ° and °).
- In Quadrant , only sin is positive.
Answer: Positive. In fact, sin(°) = sin(° − °) = sin(°) = .
Example 2 — a negative example
Is cos(°) positive or negative?
- ° falls in Quadrant (between ° and °).
- In Quadrant , only tan is positive — so cos is negative.
Answer: Negative
Practice
In which quadrant is tan positive but sin and cos both negative?
More lessons in Trigonometry · Next: Graphs of Sine and Cosine · Previous: The Unit Circle