Trigonometric Functions

1Introduction

This lecture defines trigonometric functions for every real angle as coordinates on the unit circle. Unlike a definition restricted to acute angles of right triangles, the unit-circle definition treats signs, symmetry, and periodicity uniformly.

2Definition on the Unit Circle

On the unit circle, define radian measure as signed arc length traveled along the circle from $(1,0)$(1,0). Counterclockwise length is positive, clockwise length is negative, and repeated circuits allow every real number $\theta$\theta to represent an angle. Let $P_{\theta }$P_\theta be the point reached after signed arc length $\theta$\theta. Define {cosine} and {sine} by

When $\cos \theta \ne 0$\cos\theta\ne0, define {tangent} by

The equation of the unit circle immediately gives, for every $\theta \in \mathbb{R}$\theta\in\mathbb R,

3Basic Values

Placing the ${30}^{\circ }$30^\circ-${60}^{\circ }$60^\circ-${90}^{\circ }$90^\circ and ${45}^{\circ }$45^\circ-${45}^{\circ }$45^\circ-${90}^{\circ }$90^\circ triangles in the unit circle gives:

4Symmetry and Periodicity

Reflection across the $x$x-axis and one complete rotation give

Whenever both sides are defined,

The signs in each quadrant are likewise determined by the $x$x- and $y$y-coordinates of $P_{\theta }$P_\theta.

5Connection to Rotation Matrices

Write a point at distance $r\ge 0$r\geq0 from the origin and with argument $t$t as $r{P}_t$rP_t. Define the geometric rotation $Q_{\varphi }$Q_\phi by $Q_{\varphi }\left(r{P}_t\right)=r{P}_{t+\varphi }$Q_\phi(rP_t)=rP_{t+\phi}. Addition of signed arc lengths gives $Q_{\varphi }\circ Q_{\theta }=Q_{\varphi +\theta }$Q_\phi\circ Q_\theta=Q_{\phi+\theta} independently of any trigonometric addition formula. A rotation fixes the origin and maps the parallelogram spanned by two vectors to the congruent parallelogram spanned by their images, so $Q_{\varphi }(u+v)=Q_{\varphi }u+Q_{\varphi }v$Q_\phi(u+v)=Q_\phi u+Q_\phi v. It also preserves directed distance along each line, so $Q_{\varphi }\left(cu\right)=c{Q}_{\varphi }u$Q_\phi(cu)=cQ_\phi u for every $c\in \mathbb{R}$c\in\mathbb R. Thus $Q_{\varphi }$Q_\phi is linear.

The standard basis is the pair $e_1=(1,0{)}^{\mathsf{T}}$e_1=(1,0)^\mathsf T, $e_2=(0,1{)}^{\mathsf{T}}$e_2=(0,1)^\mathsf T. A linear map preserves vector addition and multiplication by real scalars, and its matrix has the images of $e_1,e_2$e_1,e_2 as its columns. The map $Q_{\varphi }$Q_\phi is linear and preserves lengths, orthogonality, and orientation. It sends $e_1$e_1 to $u=(\cos \varphi ,\sin \varphi {)}^{\mathsf{T}}$u=(\cos\phi,\sin\phi)^\mathsf T. The image of $e_2$e_2 must be the unique unit vector perpendicular to $u$u for which $(u,v)$(u,v) has positive orientation, namely $v=(-\sin \varphi ,\cos \varphi {)}^{\mathsf{T}}$v=(-\sin\phi,\cos\phi)^\mathsf T. Hence $Q_{\varphi }$Q_\phi is represented by

Matrix multiplication represents composition of linear maps. The geometric fact that two successive rotations compose to rotation through the sum of their angles therefore gives $R_{\varphi }{R}_{\theta }=R_{\varphi +\theta }$R_\phi R_\theta=R_{\phi+\theta}. The next lecture derives the addition formulas by comparing entries.

6Advanced Topics and Prerequisites

• Euler's formula $e^{i\theta }=\cos \theta +i\sin \theta$e^{i\theta}=\cos\theta+i\sin\theta belongs after complex numbers and the complex exponential have been introduced. A proof by Taylor series also requires convergence of power series.
• Derivative formulas require the definition of a derivative, the addition formulas, and $\underset{h\to 0}{\lim }\sin h/h=1$\lim_{h\to0}\sin h/h=1.
• Inverse trigonometric functions require restricting domains so that the trigonometric functions become one-to-one, together with the concept of an inverse function.

None of these topics is a prerequisite for the unit-circle definition.

7Summary

• The values $\cos \theta$\cos\theta and $\sin \theta$\sin\theta are coordinates of a point on the unit circle.
• The fundamental identity, symmetry, and periodicity follow from the unit circle.
• Rotation matrices provide the algebraic connection to the addition formulas.

8Related Lectures