Section 5.3

The Other Trigonometric Functions

You need all six trig functions to work with identities in Chapter 7 and to solve trig equations in Section 7.5. The reciprocal and Pythagorean identities from this section are building blocks you'll use over and over. Expect at least one 'find all six trig values' problem on Exam 3.

⚡ Quick Summary

Think of sine and cosine as your two basic ingredients — the other four trig functions are just recipes made from them. Tangent is sine divided by cosine (like asking "how steep is the line?"). Cotangent flips that fraction. Secant is 1 divided by cosine, and cosecant is 1 divided by sine. That's it — no new circle, no new concepts. Everything comes from those two base ingredients.

Tangent and cotangent

\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}

Secant and cosecant

\sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta}

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

Here's the one tricky part: the names don't match the way you'd expect. Secant is the reciprocal of cosine (not sine), and cosecant is the reciprocal of sine (not cosine). A handy rule: a function and its reciprocal partner never both have the "co-" prefix. Sine pairs with cosecant, cosine pairs with secant, tangent pairs with cotangent.

Because these functions involve dividing, they sometimes "blow up" — you can't divide by zero. Tangent and secant break wherever cosine is zero, and cotangent and cosecant break wherever sine is zero. Knowing where these gaps are helps you avoid mistakes in later sections.

When you need all six trig values at some angle, start by finding sine and cosine from the unit circle, then build the other four by dividing and flipping. When simplifying a messy expression with sec, csc, tan, or cot, the go-to move is: rewrite everything in terms of sine and cosine and watch things cancel.

Key Formulas

Tangent and cotangent

\tan\theta = \frac{\sin\theta}{\cos\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}

Secant and cosecant

\sec\theta = \frac{1}{\cos\theta}, \quad \csc\theta = \frac{1}{\sin\theta}

Reciprocal pairs (product form)

\tan\theta \cdot \cot\theta = 1, \quad \sec\theta \cdot \cos\theta = 1, \quad \csc\theta \cdot \sin\theta = 1

Each pair multiplies to 1 — useful for simplifying expressions

Pythagorean identity (tangent form)

1 + \tan^2\theta = \sec^2\theta

divide $\cos^2\theta + \sin^2\theta = 1$ by $\cos^2\theta$

Pythagorean identity (cotangent form)

1 + \cot^2\theta = \csc^2\theta

divide $\cos^2\theta + \sin^2\theta = 1$ by $\sin^2\theta$

Key Takeaways

✓All six trig functions come from sin and cos: $\tan = \frac{\sin}{\cos}$ , $\sec = \frac{1}{\cos}$ , $\csc = \frac{1}{\sin}$ , $\cot = \frac{\cos}{\sin}$
✓Reciprocal pairs: sec goes with cos, csc goes with sin — the names are counterintuitive
✓Three Pythagorean identities: $\sin^2 + \cos^2 = 1$ , $1 + \tan^2 = \sec^2$ , $1 + \cot^2 = \csc^2$
✓When stuck simplifying, rewrite everything in terms of sine and cosine first

⚠️ Common Mistakes

✗Mixing up reciprocal pairs: $\sec$ goes with $\cos$ (not $\sin$ ) and $\csc$ goes with $\sin$ (not $\cos$ ) — the names are counterintuitive
✗Forgetting to rationalize denominators — e.g., writing $\frac{1}{\sqrt{3}}$ instead of $\frac{\sqrt{3}}{3}$
✗Getting the sign wrong when the angle is in QIII or QIV — always check the quadrant sign rules for each function before computing
✗Confusing $\cot\theta$ with $\frac{1}{\cos\theta}$ — cotangent is $\frac{\cos\theta}{\sin\theta}$ , not the reciprocal of cosine

Next: Section 5.4 Builds on: 5.2

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5.2 Unit Circle — Sine & Cosine

5.4 Right Triangle Trigonometry