Trig

Trigonometry

Trigonometry is named for the triangle, but there is far more to it: circles, vectors, oscillations and waves, the exponential function, and more.

Triangles – Graphs – Formulas – Radians vs degrees – Calculus – Euler’s formula – Going further

Triangles

For the right triangle shown at the left,

\(\displaystyle \sin \alpha = \frac{A}{C}\)

\(\displaystyle \cos \alpha = \frac{B}{C}\)

\(\displaystyle \tan \alpha = \frac{A}{B}\)

\(A^2 + B^2 = C^2\)

For any triangle, such as the one at the right,

\(A^2 + B^2 - 2AB \cos \theta = C^2\)

\(\displaystyle \frac{\sin \alpha}{A} = \frac{\sin \beta}{B} = \frac{\sin \theta}{C}\)

These are called the “Law of Cosines” and the “Law of Sines”, respectively.

Examples:

  1. For any triangle, you can determine all sides and angles by knowing
  2. Careful with the inverse-sine. For example, the triangle above at the right has \(B=2\), \(C=1.4\) and \(\theta=40^\circ\). If you used the Law of Sines to find angle \(\beta\), you would calculate \(\sin \beta = \frac{B}{C} \sin \theta = 0.918\). Then you would find \(\beta = \sin^{-1} 0.918 = 66.7^\circ\), but that can’t be right, since you can see that \(\beta > 90^\circ\). The problem is that \(\sin 66.7^\circ = \sin (180-66.7)^\circ\), and your calculator doesn’t give both answers. The Law of Sines is correct, with \(\beta = 113.3^\circ\).

Graphs

It is important to be familiar with the shapes of the sine and cosine graphs.

Formulas

\(\cos^2 \theta + \sin^2 \theta = 1\)

This is the most important trigonometric identity. Among many other uses, it underpins our definition of the unit vector.

\(\cos \theta = \sin \left( \theta + \frac{\pi}{2} \right)\)
\(\sin \theta = \cos \left( \theta - \frac{\pi}{2} \right)\)

A quarter-cycle phase shift can convert a sine to a cosine, and back. This should be apparent from a careful look at their graphs.

\(\cos 2 \theta = \cos^2 \theta - \sin^2 \theta\)
\(\sin 2 \theta = 2 \sin \theta \cos \theta\)

The double-angle formulas are occasionally useful. If you forget them you can easily re-derive them from Euler’s formula.

\(e^{i \theta} = \cos \theta + i \sin \theta\)

Euler’s formula is so important it has its own section below.

Radians vs degrees

Both of these angular measures are commonly used.

Further comments:

Calculus

Derivatives

These derivatives should be memorized:

\(\displaystyle \frac{d}{d\theta} \sin \theta = \cos \theta\)

\(\displaystyle \frac{d}{d\theta} \cos \theta = - \sin \theta\)

If you forget these but can sketch their graphs, you can work them out. Notice the slope of \(\sin \theta\) graph follows the \(\cos \theta\) graph. The slope of \(\cos \theta\) graph follows the opposite of the \(\sin \theta\) graph.

Don’t forget about the chain rule: \(\displaystyle \frac{d}{dt} \sin \omega t = \omega \cos \omega t\).

Integrals

These are just anti-derivatives.

\(\displaystyle \int \sin \theta = -\cos \theta +C\)

\(\displaystyle \int \cos \theta = \sin \theta +C\)

You’ll sometimes need the inverse-chain rule (also called \(u\)-substitution): \(\displaystyle \int \sin \omega t = - \frac{1}{\omega} \cos \omega t +C\).

Euler’s formula

When \(x\) is a real number, the exponential function \(e^x\) describes exponential growth and decay. When the argument is imaginary, it is made of the trig functions:

\[e^{ix} = \cos x + i \sin x\]

This is a very useful result.

In the complex plane, a number’s imaginary part is on the vertical axis and real part is on the horizontal axis (review complex numbers here). In this way, complex number \(a+ib\) is like the vector \(a \hat{\imath} + b \hat{\jmath}\). (In fact vector math was historically derived from complex math.)

In the same way, any complex number \(a+ib\) can be written in “polar form” as \(r e^{i \theta}\), where \(r=\sqrt{a^2 + b^2}\) and \(\theta\) is the angle the “vector” makes with the real axis.

Examples

Further notes

Going further

At some point I may add more about the following. Please inquire if you’re curious about them.

Last modified: October 31, 2025