Reciprocal Trig Graphs

To draw a reciprocal graph, keep the angle and change each valid output. A value of $1/2$ becomes 2; $-1/2$ becomes −2. Values 1 and −1 stay unchanged.

As sine or cosine approaches zero, its reciprocal becomes arbitrarily large in magnitude (absolute value, ignoring the sign). Vertical bars denote this: $|-3|=3$. A vertical asymptote marks that excluded angle; it is not part of the curve. The separate curve pieces are called branches; do not join them across an asymptote.

Choose a pair and move the angle. Compare the original output with its reciprocal, especially at a zero and at a turning point.

For cotangent, use $\cos\theta/\sin\theta$: zeros occur when cosine is zero, but breaks occur when sine is zero. This explains why cotangent passes through zero at $90^\circ$ and $270^\circ$.

The range is the set of possible outputs. Sine and cosine lie between −1 and 1. Their non-zero reciprocals therefore have magnitude at least 1: $y\le-1$ or $y\ge1$. Tangent and cotangent can take every real value.

The period is the smallest positive angle shift that repeats the graph. Sine, cosine, secant and cosecant repeat every $2\pi$; tangent and cotangent every $\pi$.

Secant is even: reflection in the vertical axis leaves its graph unchanged. Cosecant and cotangent are odd: a half-turn about the origin leaves their graphs unchanged. These follow from cosine being even and sine being odd.

For a sketch, mark the excluded angles first, then the points with output 1, −1 or 0 as applicable. Draw the branches through these points and repeat by the period. The graph continues beyond the displayed window.