Signs of trigonometric functions in different quadrants

Signs of trigonometric functions in different quadrants

Since $\cos\theta = x$ and $\sin\theta = y$ for the point P(x, y) on the unit circle (and $\tan\theta = y/x$), the sign of each trigonometric function in a given quadrant is decided entirely by the signs of the coordinates of P there — no separate rule is needed beyond knowing which quadrant θ's terminal ray falls in.

Quadrant I ($0 < \theta < \pi/2$): both x and y are positive, so

All trigonometric functions of θ are positive in the first quadrant.

Quadrant II ($\pi/2 < \theta < \pi$): y is positive, x is negative, so

Only $\sin\theta$ (and its reciprocal $\text{cosec}\,\theta$) is positive; $\cos\theta$ and $\tan\theta$ are negative.

Quadrant III ($\pi < \theta < 3\pi/2$): both x and y are negative, so

Only $\tan\theta$ (and $\cot\theta$) is positive; $\sin\theta$ and $\cos\theta$ are negative.

Quadrant IV ($3\pi/2 < \theta < 2\pi$): x is positive, y is negative, so

Only $\cos\theta$ (and $\sec\theta$) is positive; $\sin\theta$ and $\tan\theta$ are negative.

Remark on the reciprocal functions. Because $\text{cosec}\,\theta = 1/\sin\theta$, $\sec\theta = 1/\cos\theta$ and $\cot\theta = 1/\tan\theta$, and dividing 1 by a positive (or negative) number never changes its sign, $\text{cosec}\,\theta$ always has the same sign as $\sin\theta$, $\sec\theta$ the same sign as $\cos\theta$, and $\cot\theta$ the same sign as $\tan\theta$ (wherever they exist). So the quadrant rule above for sin, cos, tan automatically fixes the signs of all six functions. …