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Section 1.3 Trigonometric Functions

Definition 1.3.1.

The unit circle is the circle centered at the origin of radius 1. Its equation is \(x^2+y^2=1\text{.}\) With an angle in standard position, the measure of an angle in radians is the length of the corresponding arc.

Note that for a circle of radius \(r\text{,}\) the length of an arc corresponding to an angle \(\theta\) is \(r\theta\text{.}\)

Table 1.3.2. Common angles, measured in radians
\(2\pi\) full circle
\(\pi\) half circle
\(\frac{\pi}{2}\) quarter circle/right angle
\(\frac{\pi}{3}\) bigger than \(\frac{\pi}{4}\text{,}\) \(60^{\circ}\)
\(\frac{\pi}{4}\) half a right angle, \(45^{\circ}\)
\(\frac{\pi}{6}\) smaller than \(\frac{\pi}{4}\text{,}\) \(30^{\circ}\)

Angles larger than \(2\pi\) wrap around the circle, e.g., \(\frac{17\pi}{6}\) is equivalent to \(\frac{5\pi}{6}\text{.}\)

Negative angles are positioned clockwise.

Two points of view of sine and cosine:

  • Right triangles

  • Unit circle

The unit circle
Figure 1.3.3. Unit circle

Based on the unit circle, we immediately know the values for \(\theta=0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\text{.}\)

The unit circle also makes the signs of sine and cosine clear for each of the four quadrants.

Table 1.3.4. The rule of 1-2-3
\(\frac{\pi}{6}\) \(\frac{\pi}{4}\) \(\frac{\pi}{3}\)
\(\sin\) \(\frac{1}{2}\) \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{3}}{2}\)
\(\cos\) \(\frac{\sqrt{3}}{2}\) \(\frac{\sqrt{2}}{2}\) \(\frac{1}{2}\)
\(\tan=\frac{\sin}{\cos}\) \(\frac{1}{\sqrt{3}}\) \(1\) \(\sqrt{3}\)

Note 1.3.5.

MyMathLab may sometimes require rational denominators, e.g., \(\frac{\sqrt{3}}{3}\) instead of \(\frac{1}{\sqrt{3}}\) and \(\sqrt{2}\) rather than \(\frac{2}{\sqrt{2}}\text{.}\)

Table 1.3.6. The six basic trig functions
\(\sin\theta\) \(\csc\theta=\frac{1}{\sin\theta}\)
\(\cos\theta\) \(\sec\theta=\frac{1}{\cos\theta}\)
\(\tan\theta=\frac{\sin\theta}{\cos\theta}\) \(\cot\theta=\frac{1}{\tan\theta}=\frac{\cos\theta}{\sin\theta}\)

Example 1.3.7.

Compute the following:

  • \(\displaystyle \sin\left(\frac{5\pi}{3}\right)\)

  • \(\displaystyle \sec\left(\frac{-3\pi}{4}\right)\)

  • \(\displaystyle \cot 3\pi\)

  • If you know that \(\tan\theta = \frac{-5}{6}\) and that \(\theta \in \left[\frac{\pi}{2},\pi\right]\text{,}\) then what are \(\sin\theta\) and \(\cos\theta\text{?}\)

  • If you know that \(\cos\theta = \frac{2}{5}\) and that \(\theta \in \left[-\pi,-2\pi\right]\text{,}\) then what are \(\sin\theta\) and \(\tan\theta\text{?}\)

Answer.

  • \(\displaystyle \frac{-\sqrt{3}}{2}\)

  • \(\displaystyle \frac{-2}{\sqrt{2}}=-\sqrt{2}\)

  • \(\displaystyle 0\)

  • \(\sin\theta = \frac{5}{\sqrt{61}}\text{;}\) \(\cos\theta = \frac{-6}{\sqrt{61}}\)

  • \(\sin\theta = -\frac{\sqrt{21}}{5}\text{;}\) \(\tan\theta = -\frac{\sqrt{21}}{2}\)

Definition 1.3.8. Periodic function.

A function \(f\) is periodic if there is a positive number \(p\) such that \(f(x+p)=f(x)\) for all \(x\text{.}\) The smallest such \(p\) is called the period.

Example 1.3.10.

Which of the trig functions are even and which are odd?

Answer.

\(\cos\) and \(\sec\) are even, while the others are all odd.

Definition 1.3.11. Sinusoid.

The sinusoid or general sine function is given by the formula

\begin{equation*} f(x) = A \sin\left(\frac{2\pi}{B}(x-C)\right) + D \end{equation*}

where

  • \(|A|\) is the amplitude

  • \(|B|\) is the period

  • \(C\) is the horizontal shift

  • \(D\) is the vertical shift

The sinusoid illustrated
Figure 1.3.12. Sinusoid

Example 1.3.13.

Find the period and graph

  • \(\displaystyle y=\sin(3x)\)

  • \(\displaystyle y=-3\cos\left(\frac{\pi}{5} x\right) + 1\)

Solution.

There are a variety of ways to approach finding the period. One of the more straightforward is to set the coefficient of the variable equal to \(\frac{2\pi}{B}\) and solve for \(B\text{.}\)

  • Solving \(3=\frac{2\pi}{B}\text{,}\) we get the period is \(\frac{3}{2\pi}\text{.}\)

  • Solving \(\frac{\pi}{5} = \frac{2\pi}{B}\text{,}\) the \(\pi\)s cancel and the period is \(\frac52\text{.}\)

Graphs of sinusoids
Figure 1.3.14. Graphs of two sinusoids: \(y=\sin(3x)\) and \(y=-3\cos\left(\frac{\pi}{5} x\right) + 1\text{.}\)

Pythagorean Identities:

\begin{align*} \sin^2 \theta + \cos^2 \theta \amp = 1\\ \tan^2 \theta + 1 \amp = \sec^2\theta\\ 1 + \cot^2\theta \amp = \csc^2\theta \end{align*}

Addition Formulas:

\begin{align*} \cos(A+B)\amp =\cos(A)\cos(B)-\sin(A)\sin(B)\\ \sin(A+B)\amp =\cos(A)\sin(B)+\cos(B)\sin(A) \end{align*}

Double-Angle Formulas

\begin{align*} \cos 2\theta \amp = \cos^2\theta - \sin^2\theta\\ \sin 2\theta \amp = 2\sin\theta \cos\theta \end{align*}

Half-Angle Formulas

\begin{align*} \cos^2\theta \amp = \frac{1+\cos 2\theta}{2}\\ \sin^2 \theta \amp = \frac{1-\cos 2\theta}{2} \end{align*}

Law of Cosines

\begin{equation*} c^2=a^2+b^2-2ab\cos\theta \end{equation*}

Law of Sines

\begin{equation*} \frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c} \end{equation*}

Useful inequalities

\begin{gather*} -|\theta| \leq \sin\theta \leq |\theta|\\ -|\theta| \leq 1-\cos\theta \leq |\theta| \end{gather*}

Example 1.3.15.

Compute the values of \(\sin \frac{7\pi}{12}\) and \(\cos \frac{\pi}{8}\text{.}\)