The Unit Circle

The unit circle is the circle of radius 1 centered at \((0,0)\). The equation of the unit circle is \(x^{2}+y^{2}=1\).

Terminal Points on the Unit Circle

The terminal point \(P(x, y)\) determined by the real number \(t\) is the point obtained by traveling counterclockwise a distance \(t\) along the unit circle, starting at \((1,0)\).

Special terminal points are listed in Table 5.1.1.

The Reference Number

The reference number associated with the real number \(t\) is the shortest distance along the unit circle between the terminal point determined by \(t\) and the \(x\)-axis.

The Trigonometric Functions

Let \(P(x, y)\) be the terminal point on the unit circle determined by the real number \(t\). Then for nonzero values of the denominator the trigonometric functions are defined as follows.

\[ \begin{array}{lll} \sin t=y & \cos t=x & \tan t=\frac{y}{x} \\ \csc t=\frac{1}{y} & \sec t=\frac{1}{x} & \cot t=\frac{x}{y} \end{array} \]

Special Values of the Trigonometric Functions

The trigonometric functions have the following values at the special values of \(t\).

\(\boldsymbol{t}\) \(\boldsymbol{\operatorname { s i n } t}\) \(\boldsymbol{\operatorname { c o s }} \boldsymbol{t}\) \(\boldsymbol{\operatorname { t a n }} \boldsymbol{t}\) \(\boldsymbol{\operatorname { c s c }} \boldsymbol{t}\) \(\sec t\) \(\boldsymbol{\operatorname { c o t }} \boldsymbol{t}\)
0 0 1 0 - 1 -
\(\frac{\pi}{6}\) \(\frac{1}{2}\) \(\frac{\sqrt{3}}{2}\) \(\frac{\sqrt{3}}{3}\) 2 \(\frac{2 \sqrt{3}}{3}\) \(\sqrt{3}\)
\(\frac{\pi}{4}\) \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{2}}{2}\) 1 \(\sqrt{2}\) \(\sqrt{2}\) 1
\(\frac{\pi}{3}\) \(\frac{\sqrt{3}}{2}\) \(\frac{1}{2}\) \(\sqrt{3}\) \(\frac{2 \sqrt{3}}{3}\) 2 \(\frac{\sqrt{3}}{3}\)
\(\frac{\pi}{2}\) 1 0 - 1 - 0
\(\pi\) 0 -1 0 - -1 -
\(\frac{3 \pi}{2}\) -1 0 - -1 - 0

Basic Trigonometric Identities

An identity is an equation that is true for all values of the variable. The basic trigonometric identities are as follows.

Reciprocal Identities

\[ \csc t=\frac{1}{\sin t} \quad \sec t=\frac{1}{\cos t} \quad \cot t=\frac{1}{\tan t} \]

Pythagorean Identities

\[ \begin{aligned} \sin ^{2} t+\cos ^{2} t & =1 \\ \tan ^{2} t+1 & =\sec ^{2} t \\ 1+\cot ^{2} t & =\csc ^{2} t \end{aligned} \]

Even-Odd Properties

\[ \begin{array}{lll} \sin (-t)=-\sin t & \cos (-t)=\cos t & \tan (-t)=-\tan t \\ \csc (-t)=-\csc t & \sec (-t)=\sec t & \cot (-t)=-\cot t \end{array} \]

Periodic Properties

A function \(f\) is periodic if there is a positive number \(p\) such that \(f(x+p)=f(x)\) for every \(x\). The least such \(p\) is called the period of \(f\). The sine and cosine functions have period \(2 \pi\), and the tangent function has period \(\pi\).

\[ \begin{aligned} \sin (t+2 \pi) & =\sin t \\ \cos (t+2 \pi) & =\cos t \\ \tan (t+\pi) & =\tan t \end{aligned} \]

Graphs of the Sine and Cosine Functions

The graphs of sine and cosine have amplitude 1 and period \(2 \pi\).

Amplitude 1, Period \(2 \pi\)

Graphs of Transformations of Sine and Cosine | Section 5.3

An appropriate interval on which to graph one complete period is \([b, b+(2 \pi / k)]\).

Graphs of the Tangent and Cotangent Functions

These functions have period \(\pi\).

\(y=\cot x\)

To graph one period of \(y=a \tan k x\), an appropriate interval is \(\left(-\frac{\pi}{2 k}, \frac{\pi}{2 k}\right)\). To graph one period of \(y=a \cot k x\), an appropriate interval is \(\left(0, \frac{\pi}{k}\right)\).

Graphs of the Secant and Cosecant Functions

These functions have period \(2 \pi\).

\[ y=\sec x \]

To graph one period of \(y=a \sec k x\) or \(y=a \csc k x\), an appropriate interval is \(\left(0, \frac{2 \pi}{k}\right)\).

Inverse Trigonometric Functions

Inverse functions of the trigonometric functions have the following domain and range.

Function Domain Range
\(\sin ^{-1}\) \([-1,1]\) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)
\(\cos ^{-1}\) \([-1,1]\) \([0, \pi]\)
\(\tan ^{-1}\) \((-\infty, \infty)\) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

The inverse trigonometric functions are defined as follows.

\[ \begin{aligned} \sin ^{-1} x=y & \Leftrightarrow \quad \sin y=x \\ \cos ^{-1} x=y & \Leftrightarrow \quad \cos y=x \\ \tan ^{-1} x=y & \Leftrightarrow \quad \tan y=x \end{aligned} \]

Graphs of these inverse functions are shown below.

Harmonic Motion

An object is in simple harmonic motion if its displacement \(y\) at time \(t\) is modeled by \(y=a \sin \omega t\) or \(y=a \cos \omega t\). In this case the amplitude is \(|a|\), the period is \(2 \pi / \omega\), and the frequency is \(\omega /(2 \pi)\).

Damped Harmonic Motion

An object is in damped harmonic motion if its displacement \(y\) at time \(t\) is modeled by \(y=k e^{-c t} \sin \omega t\) or \(y=k e^{-c t} \cos \omega t, c>0\). In this case \(c\) is the damping constant, \(k\) is the initial amplitude, and \(2 \pi / \omega\) is the period.

Phase

Any sine curve can be expressed in the following equivalent forms:

\[ \begin{array}{ll} y=A \sin (k t-b), & \text { the phase is } b \\ y=A \sin k\left(t-\frac{b}{k}\right), & \text { the horizontal shift is } \frac{b}{k} \end{array} \]

The phase (or phase angle) \(b\) is the initial angular position of the motion. The number \(b / k\) is also called the lag time \((b>0)\) or lead time \((b<0)\).

Suppose that two objects are in harmonic motion with the same period modeled by

\[ y_{1}=A \sin (k t-b) \quad \text { and } \quad y_{2}=A \sin (k t-c) \]

The phase difference between \(y_{1}\) and \(y_{2}\) is \(b-c\). The motions are "in phase" if the phase difference is a multiple of \(2 \pi\); otherwise, the motions are "out of phase."