Trigonometry / Unit 3: The Unit Circle and Trig Functions of Any Angle
3.2 Signs of the Trig Functions
Notes
The sign of each function depends on the signs of $x$ and $y$ in the quadrant.
- Quadrant I: all positive. Quadrant II: only sine (and cosecant) positive.
- Quadrant III: only tangent (and cotangent) positive. Quadrant IV: only cosine (and secant) positive.
- Memory aid: All Students Take Calculus (I, II, III, IV).
- Reciprocal functions have the same sign as the original.
Practice
In which quadrant is $\sin\theta > 0$ and $\cos\theta < 0$?
Sine is $\displaystyle \frac{y}{r}$ (positive when $y > 0$) and cosine is $\displaystyle \frac{x}{r}$ (negative when $x < 0$). That is Quadrant II.
If $\sin\theta < 0$ and $\tan\theta > 0$, in which quadrant is $\theta$?
Both $x$ and $y$ are negative, so sine is negative and tangent is positive.
Which value is positive?
$100^\circ$ is in Quadrant II, where sine is positive. $300^\circ$ is in Quadrant IV, where sine is negative.
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