5.6 Secant, cosecant, cotangent and the inverse trigonometric functions — Learn

Prerequisites: the graphs of sine, cosine and tangent; the identity $\sin^2\theta + \cos^2\theta = 1$; finding an angle from a known ratio.

By the end of this section you can: define secant, cosecant and cotangent; use the identities that link them to tangent; name the domain restrictions; find an angle with an inverse trigonometric function and give the principal value.

1. Three new functions

Each of the three new functions is the reciprocal of one you already know.

The names come from the geometry of a circle, but the definitions above are all you need.

Each function is undefined where its denominator is zero:

• $\sec\theta$ is undefined where $\cos\theta = 0$, that is at $\theta = \dfrac{\pi}{2}$, $\dfrac{3\pi}{2}$ and so on;
• $\operatorname{cosec}\theta$ is undefined where $\sin\theta = 0$, that is at multiples of $\pi$;
• $\cot\theta$ is undefined where $\tan\theta = 0$, also at multiples of $\pi$.

The graph of $y = \sec x$ has a minimum value of $1$ and a maximum value of $-1$, with branches that curve away towards the asymptotes. It never takes a value between $-1$ and $1$.

2. Two identities

Divide $\sin^2\theta + \cos^2\theta = 1$ by $\cos^2\theta$:

Divide the same identity by $\sin^2\theta$:

These two identities are the ones to reach for when a question mixes a reciprocal function with a squared term.

Example 1. Simplify $\sec^2\theta - \tan^2\theta$.

From the first identity, $\sec^2\theta = 1 + \tan^2\theta$. Therefore

Example 2. Given that $\sin\theta = \dfrac{3}{5}$ and $\theta$ is acute, find $\cos\theta$, $\sec\theta$, $\operatorname{cosec}\theta$ and $\cot\theta$.

Use $\cos^2\theta = 1 - \sin^2\theta = 1 - \dfrac{9}{25} = \dfrac{16}{25}$. Since $\theta$ is acute, $\cos\theta$ is positive:

Then

3. Inverse trigonometric functions

The inverse function of sine is written $\arcsin$, or $\sin^{-1}$. It answers the question "which angle has this sine?". The output is an angle.

Each inverse function has a principal range, the interval of angles it returns:

The principal range restricts the answer to one angle. Section 5.10 shows how to find all the other angles that share the same value.

Watch out. Three mistakes are common with these functions.

Reading $\sin^{-1}$ as a reciprocal. The notation $\sin^{-1}\theta$ means $\arcsin\theta$, the inverse function, not $\dfrac{1}{\sin\theta}$. The reciprocal is $\operatorname{cosec}\theta$. Write $\arcsin$ when you mean the inverse, and the confusion disappears.

Forgetting the domain. The cosecant, secant and cotangent have holes in their domains. At $\theta = 0$, for example, $\operatorname{cosec}\theta$ and $\cot\theta$ are both undefined, because $\sin\theta = 0$.

Losing the sign when you find cosine. From $\sin^2\theta + \cos^2\theta = 1$ you get $\cos\theta = \pm\sqrt{1 - \sin^2\theta}$. The quadrant decides the sign. Re-check: if the question says acute, take the positive root; if it names another quadrant, test the sign against the CAST table.

You should now be able to

• define secant, cosecant and cotangent as reciprocals;
• state where each is undefined;
• use $\sec^2\theta = 1 + \tan^2\theta$ and $\operatorname{cosec}^2\theta = 1 + \cot^2\theta$;
• find the other ratios from one known ratio and the quadrant;
• use $\arcsin$, $\arccos$ and $\arctan$ and give the principal value;
• explain the difference between $\sin^{-1}\theta$ and $(\sin\theta)^{-1}$.