What Are Secant, Cosecant, and Cotangent?

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These definitions apply to angles on the unit circle, not only to acute angles in right-angled triangles. Their signs therefore depend on the quadrant, just as the signs of sine, cosine, and tangent do.

For example, if $\cos \theta = \frac{1}{2}$, then

If instead $\sin \theta=-\frac{3}{5}$, then $\cosec \theta=-\frac{5}{3}$. Taking a reciprocal does not remove the original sign.

Because reciprocal ratios reverse magnitudes, a primary ratio close to zero produces a reciprocal ratio with a large absolute value. For example, if $\cos \theta=\frac{1}{10}$, then $\sec \theta=10$. This behaviour explains why the secant graph grows rapidly near angles where cosine is zero and why its branches approach vertical asymptotes without crossing them within a continuous branch.

A reciprocal ratio is undefined whenever its denominator equals zero. Therefore, $\sec \theta$ is undefined when $\cos \theta=0$. Both $\cosec \theta$ and $\cot \theta$ are undefined when $\sin \theta=0$. These excluded values correspond to vertical asymptotes on the relevant graphs.

The reciprocal definitions also produce the Pythagorean identities in AHL 3.9. Starting from $\sin^2\theta+\cos^2\theta=1$, divide every term by $\cos^2\theta$:

Dividing every term instead by $\sin^2\theta$ gives

These identities allow expressions containing reciprocal ratios to be simplified or rewritten. For instance, $\sec^2\theta-\tan^2\theta=1$, provided all ratios involved are defined.

A common misconception is that reciprocal functions and inverse functions are identical. They are not: $\cosec\theta=\frac{1}{\sin\theta}$, whereas $\sin^{-1}x=\arcsin x$ represents the angle whose sine is $x$. Similarly, $\sec\theta$ is not $\cos^{-1}\theta$.

Exam technique: This is IB Mathematics AA HL content from AHL 3.9, relevant to Papers 1 and 2. Show the reciprocal substitution before simplifying, use exact values where possible on Paper 1, apply the identities in either direction, and reject any value that makes a denominator zero.