Chapter 2: Trigonometric Functions
About This Chapter
In Chapter 1, trigonometric ratios were defined for $0° < \theta < 90°$. In this chapter, we extend the domain of angles to all real numbers and define trigonometric functions. We also introduce radian measure, a new unit for angles, and study the graphs and periodicity of trigonometric functions.
General Angles
Angles as Rotations
Previously, angles were considered only in the range $0°$ to $360°$. We now extend this to general angles that include the number of full rotations.
Definition 2.1 (General Angle)
The amount of counterclockwise rotation of a ray about the origin is called a positive angle, and the amount of clockwise rotation is called a negative angle. One full rotation equals $360°$, and $n$ full rotations correspond to an angle of $360n°$.
For example:
- $390° = 360° + 30°$ (one full rotation plus $30°$)
- $-45°$ ($45°$ clockwise)
- $720°$ (two full rotations)
Figure 1: Examples of general angles
Quadrants
The coordinate plane is divided into four regions by the $x$-axis and the $y$-axis. These regions are called quadrants.
Figure 2: The four quadrants of the coordinate plane
Radian Measure
Besides degree measure (°), there is another unit for angles called radian measure.
Definition 2.2 (Radian)
In a circle of radius $r$, the central angle subtended by an arc of length $r$ is defined as $1$ radian ($1$ rad).
Figure 3: Definition of 1 radian
Relationship Between Degrees and Radians
The circumference of a circle of radius $r$ is $2\pi r$. Therefore, one full revolution ($360°$) corresponds to $2\pi$ radians.
Definition 2.3 (Degree–Radian Conversion)
\begin{align} 360° &= 2\pi \text{ rad} \label{eq:360-2pi}\\ 180° &= \pi \text{ rad} \label{eq:180-pi}\\ 1° &= \frac{\pi}{180} \text{ rad} \label{eq:1deg-rad}\\ 1 \text{ rad} &= \frac{180°}{\pi} \approx 57.3° \label{eq:1rad-deg} \end{align}Common Angle Correspondences
Figure 4: Correspondence between degrees and radians
Why Use Radian Measure?
In mathematics and physics, radian measure is more convenient than degree measure. In differentiation (computing rates of change), radian measure yields simpler formulas.
In radian measure, the following elegant formulas hold:
$$\frac{d}{dx}\sin x = \cos x, \quad \frac{d}{dx}\cos x = -\sin x$$In contrast, using degree measure introduces an extra factor:
$$\frac{d}{dx}\sin x = \frac{\pi}{180}\cos x, \quad \frac{d}{dx}\cos x = -\frac{\pi}{180}\sin x$$Thus, radian measure yields the most natural and concise form of mathematical formulas. This is why radians are the standard in mathematics and physics.
Definition via the Unit Circle
To define trigonometric functions for general angles, we use the unit circle (a circle of radius 1).
Definition 2.4 (Trigonometric Functions via the Unit Circle)
On the coordinate plane, consider the circle of radius 1 centered at the origin (the unit circle). Let $P(\cos\theta, \sin\theta)$ be the point where the ray rotated counterclockwise by angle $\theta$ from the positive $x$-axis intersects the unit circle.
\begin{align} \cos\theta &= \text{the } x\text{-coordinate of } P \label{eq:cos-unit-circle}\\ \sin\theta &= \text{the } y\text{-coordinate of } P \label{eq:sin-unit-circle}\\ \tan\theta &= \frac{\sin\theta}{\cos\theta} \quad (\cos\theta \neq 0) \label{eq:tan-unit-circle} \end{align}Figure 5: Trigonometric functions defined via the unit circle
This definition extends trigonometric functions to any angle $\theta$, including negative angles and angles greater than $360°$.
Graphs of Trigonometric Functions
Graph of y = sin x
Figure 6: Graph of y = sin x
- Period: $2\pi$
- Range: $-1 \leq \sin x \leq 1$
- Maximum value $1$ at $x = \dfrac{\pi}{2} + 2n\pi$
- Minimum value $-1$ at $x = \dfrac{3\pi}{2} + 2n\pi$
- Zero at $x = n\pi$
- Sign: positive in Quadrants I and II ($0 < x < \pi$), negative in Quadrants III and IV ($\pi < x < 2\pi$)
Graph of y = cos x
Figure 7: Graph of y = cos x
- Period: $2\pi$
- Range: $-1 \leq \cos x \leq 1$
- Maximum value $1$ at $x = 2n\pi$
- Minimum value $-1$ at $x = \pi + 2n\pi$
- Zero at $x = \dfrac{\pi}{2} + n\pi$
- Sign: positive in Quadrants I and IV ($-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}$), negative in Quadrants II and III ($\dfrac{\pi}{2} < x < \dfrac{3\pi}{2}$)
Graph of y = tan x
Figure 8: Graph of y = tan x
- Period: $\pi$
- Range: all real numbers
- Undefined at $x = \dfrac{\pi}{2} + n\pi$ (vertical asymptotes)
* A line that a graph approaches but never crosses is called an asymptote. The dashed lines in Figure 8 are the asymptotes. - Zero at $x = n\pi$
- Sign: positive in Quadrants I and III, negative in Quadrants II and IV
Properties of Trigonometric Functions
Periodicity
Since going once around the unit circle returns to the starting point, trigonometric functions are periodic functions.
Theorem 2.1 (Periodicity)
\begin{align} \sin(\theta + 2\pi) &= \sin\theta \label{eq:sin-period}\\ \cos(\theta + 2\pi) &= \cos\theta \label{eq:cos-period}\\ \tan(\theta + \pi) &= \tan\theta \label{eq:tan-period} \end{align}The period of sin and cos is $2\pi$, and the period of tan is $\pi$.
Even and Odd Functions
Theorem 2.2 (Parity)
\begin{align} \sin(-\theta) &= -\sin\theta \quad \text{(odd function)} \label{eq:sin-odd}\\ \cos(-\theta) &= \cos\theta \quad \text{(even function)} \label{eq:cos-even}\\ \tan(-\theta) &= -\tan\theta \quad \text{(odd function)} \label{eq:tan-odd} \end{align}Worked Examples
Example 1: Converting Between Degrees and Radians
Convert the following angles.
(1) Convert $135°$ to radians.
(2) Convert $\dfrac{5\pi}{6}$ radians to degrees.
Solution
(1) Using the relation $180° = \pi$ rad:
$$135° = 135 \times \frac{\pi}{180} = \frac{135\pi}{180} = \frac{3\pi}{4}$$(2) Using the relation $1 \text{ rad} = \dfrac{180°}{\pi}$:
$$\frac{5\pi}{6} \text{ rad} = \frac{5\pi}{6} \times \frac{180°}{\pi} = \frac{5 \times 180°}{6} = 150°$$(1) $135° = \dfrac{3\pi}{4}$ rad (2) $\dfrac{5\pi}{6}$ rad $= 150°$
Example 2: Coordinates on the Unit Circle and Trigonometric Values
Find $\sin\theta$, $\cos\theta$, and $\tan\theta$ when $\theta = \dfrac{2\pi}{3}$ ($= 120°$).
Solution
$\theta = \dfrac{2\pi}{3}$ is a second-quadrant angle. The reference angle (the acute angle with the $x$-axis) is $\pi - \dfrac{2\pi}{3} = \dfrac{\pi}{3}$ ($= 60°$).
From Chapter 1, $\sin 60° = \dfrac{\sqrt{3}}{2}$ and $\cos 60° = \dfrac{1}{2}$.
In Quadrant II, $\sin > 0$ and $\cos < 0$, so:
\begin{align} \sin\frac{2\pi}{3} &= \sin 60° = \frac{\sqrt{3}}{2} \\ \cos\frac{2\pi}{3} &= -\cos 60° = -\frac{1}{2} \\ \tan\frac{2\pi}{3} &= \frac{\sin\frac{2\pi}{3}}{\cos\frac{2\pi}{3}} = \frac{\frac{\sqrt{3}}{2}}{-\frac{1}{2}} = -\sqrt{3} \end{align}$\sin\dfrac{2\pi}{3} = \dfrac{\sqrt{3}}{2}$, $\cos\dfrac{2\pi}{3} = -\dfrac{1}{2}$, $\tan\dfrac{2\pi}{3} = -\sqrt{3}$
Example 3: Period and Amplitude of a Trigonometric Graph
Find the period and amplitude of $y = 3\sin 2x$.
Solution
For a function of the form $y = A\sin Bx$:
- Amplitude $= |A|$
- Period $= \dfrac{2\pi}{|B|}$
For $y = 3\sin 2x$, we have $A = 3$ and $B = 2$:
\begin{align} \text{Amplitude} &= |3| = 3 \\ \text{Period} &= \frac{2\pi}{|2|} = \pi \end{align}Amplitude $3$, Period $\pi$
Practice Problems
Problem 1
Convert the following angles to radians.
(1) $210°$ (2) $315°$ (3) $-60°$
Hint
Use the relation $180° = \pi$ rad. To convert degrees to radians, multiply by $\dfrac{\pi}{180}$. Negative angles can be converted in the same way.
Solution
(1) $210° = 210 \times \dfrac{\pi}{180} = \dfrac{7\pi}{6}$
(2) $315° = 315 \times \dfrac{\pi}{180} = \dfrac{7\pi}{4}$
(3) $-60° = -60 \times \dfrac{\pi}{180} = -\dfrac{\pi}{3}$
Problem 2
When $\theta = \dfrac{5\pi}{4}$, find the coordinates of the point $P$ on the unit circle and determine $\sin\theta$, $\cos\theta$, and $\tan\theta$.
Hint
Since $\dfrac{5\pi}{4} = \pi + \dfrac{\pi}{4}$, the angle lies in Quadrant III and the reference angle is $\dfrac{\pi}{4}$ ($= 45°$). Note that in Quadrant III, both $\sin$ and $\cos$ are negative.
Solution
The reference angle is $\dfrac{\pi}{4}$ ($45°$), and $\sin 45° = \cos 45° = \dfrac{\sqrt{2}}{2}$.
In Quadrant III, $\sin < 0$ and $\cos < 0$, so:
$P\!\left(-\dfrac{\sqrt{2}}{2},\; -\dfrac{\sqrt{2}}{2}\right)$
$\sin\dfrac{5\pi}{4} = -\dfrac{\sqrt{2}}{2}$, $\cos\dfrac{5\pi}{4} = -\dfrac{\sqrt{2}}{2}$, $\tan\dfrac{5\pi}{4} = \dfrac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1$
Problem 3
Find all values of $\theta$ satisfying $\sin\theta = -\dfrac{1}{2}$ in the range $0 \leq \theta < 2\pi$.
Hint
Start from the angle where $\sin\theta = \dfrac{1}{2}$ (which is $\dfrac{\pi}{6}$). Sine is negative in Quadrants III and IV.
Solution
Since $\sin\dfrac{\pi}{6} = \dfrac{1}{2}$, the reference angle is $\dfrac{\pi}{6}$.
$\sin\theta < 0$ in Quadrants III and IV:
Quadrant III: $\theta = \pi + \dfrac{\pi}{6} = \dfrac{7\pi}{6}$
Quadrant IV: $\theta = 2\pi - \dfrac{\pi}{6} = \dfrac{11\pi}{6}$
Therefore $\theta = \dfrac{7\pi}{6},\; \dfrac{11\pi}{6}$
Problem 4
Find the period, amplitude, maximum value, and minimum value of $y = 2\cos 3x$.
Hint
For $y = A\cos Bx$, the amplitude is $|A|$ and the period is $\dfrac{2\pi}{|B|}$. The range of $\cos$ is $-1 \leq \cos 3x \leq 1$.
Solution
With $A = 2$ and $B = 3$:
Amplitude $= |2| = 2$
Period $= \dfrac{2\pi}{|3|} = \dfrac{2\pi}{3}$
From $-1 \leq \cos 3x \leq 1$, multiplying by $2$: $-2 \leq 2\cos 3x \leq 2$
Maximum value: $2$ (when $\cos 3x = 1$, i.e., $x = \dfrac{2n\pi}{3}$)
Minimum value: $-2$ (when $\cos 3x = -1$, i.e., $x = \dfrac{(2n+1)\pi}{3}$)