Trigonometric Functions

Overview

Trigonometric functions take an angle as input and output a number that describes a ratio in a right triangle or a coordinate on the unit circle. In earlier chapters you met the basic trigonometric ratios (sine, cosine, tangent) and the unit circle. Here we treat these ratios as full-fledged functions: we think about their domains, ranges, basic shapes, and how they behave as you vary the input.

In this chapter we will focus on:

• Seeing sine, cosine, and tangent as functions.
• Describing and using their basic graphs.
• Understanding their key features: period, amplitude, vertical shifts, and phase shifts.
• Recognizing and sketching simple transformations of trig functions.

The detailed work with exact values and the unit circle, and specific identities, is handled in their own chapters; here we treat those results as tools we can use when needed.

Trigonometric Ratios as Functions

A trigonometric function is a rule that assigns a real number to each angle (where it is defined). We usually measure the input angle in radians for calculus and advanced work, but degrees can be used as well. Unless explicitly stated, you should assume angles are in radians.

The three primary trigonometric functions are:

• Sine function: $y = \sin x$.
• Cosine function: $y = \cos x$.
• Tangent function: $y = \tan x$.

Here $x$ is the angle. On the unit circle:

• $\sin x$ is the $y$-coordinate of the point on the circle.
• $\cos x$ is the $x$-coordinate.
• $\tan x = \dfrac{\sin x}{\cos x}$, so it is the ratio of the $y$-coordinate to the $x$-coordinate (when $\cos x \neq 0$).

From the unit circle picture, these functions are inherently periodic: their values repeat as you go around the circle.

Sine and Cosine as Wave Functions

Domain and Range

For the basic sine and cosine functions:

• Domain: all real numbers
  $$\text{Domain}(\sin x) = \text{Domain}(\cos x) = (-\infty, \infty).$$
• Range: values between $-1$ and $1$
  $$\text{Range}(\sin x) = \text{Range}(\cos x) = [-1, 1].$$

They never go above $1$ or below $-1$ because on the unit circle the coordinates are always between $-1$ and $1$.

Periodicity

A function is periodic if its values repeat at regular intervals. For sine and cosine:

• Fundamental period: $2\pi$ (in radians).
• This means:
  $$\sin(x + 2\pi) = \sin x \quad\text{for all } x,$$
  $$\cos(x + 2\pi) = \cos x \quad\text{for all } x.$$

You also have simple symmetries (using identities):

• $\sin(-x) = -\sin x$ (sine is an odd function: symmetric about the origin).
• $\cos(-x) = \cos x$ (cosine is an even function: symmetric about the $y$-axis).

Key Values

Using the unit circle chapter, certain “special angle” values are especially useful when working with graphs:

• $\sin 0 = 0$, $\cos 0 = 1$.
• $\sin \dfrac{\pi}{2} = 1$, $\cos \dfrac{\pi}{2} = 0$.
• $\sin \pi = 0$, $\cos \pi = -1$.
• $\sin \dfrac{3\pi}{2} = -1$, $\cos \dfrac{3\pi}{2} = 0$.
• $\sin 2\pi = 0$, $\cos 2\pi = 1$.

These points form the “skeleton” for sketching $\sin x$ and $\cos x$.

Basic Graph of $y = \sin x$

Think of $y = \sin x$ as a smooth, repeating wave:

• It crosses the $x$-axis at $x = 0, \pi, 2\pi, 3\pi, \dots$.
• It reaches maximum $y = 1$ at $x = \dfrac{\pi}{2} + 2k\pi$.
• It reaches minimum $y = -1$ at $x = \dfrac{3\pi}{2} + 2k\pi$.
• One full cycle runs from $x = 0$ to $x = 2\pi$.

If you list key points for one period:

• $(0, 0)$
• $\left(\dfrac{\pi}{2}, 1\right)$
• $(\pi, 0)$
• $\left(\dfrac{3\pi}{2}, -1\right)$
• $(2\pi, 0)$

Then repeat that shape every $2\pi$ units.

Basic Graph of $y = \cos x$

The cosine graph is also a smooth, repeating wave, but “shifted” relative to sine:

• It starts at its maximum: $(0, 1)$.
• It crosses the $x$-axis at $x = \dfrac{\pi}{2}, \dfrac{3\pi}{2}, \dots$.
• It reaches minimum $y = -1$ at $x = \pi + 2k\pi$.
• One full cycle from $x = 0$ to $x = 2\pi$.

Key points for one period:

• $(0, 1)$
• $\left(\dfrac{\pi}{2}, 0\right)$
• $(\pi, -1)$
• $\left(\dfrac{3\pi}{2}, 0\right)$
• $(2\pi, 1)$

Notice that $\cos x$ is just a horizontal shift of $\sin x$:
$$\cos x = \sin\left(x + \dfrac{\pi}{2}\right).$$

Vertical Stretch and Amplitude

The general forms
$$y = a \sin x, \quad y = a \cos x$$
stretch or shrink the graph vertically:

• If $|a| > 1$, the graph is “taller” (stretched).
• If $0 < |a| < 1$, the graph is “shorter” (compressed).
• If $a < 0$, the graph is flipped across the $x$-axis (reflected).

For $y = a\sin x$ and $y = a\cos x$:

• Amplitude is $|a|$.
• Range becomes $[-|a|, |a|]$.

Examples:

• $y = 2 \sin x$ has amplitude $2$ and range $[-2, 2]$.
• $y = -\dfrac{1}{2} \cos x$ has amplitude $\dfrac{1}{2}$ and is flipped vertically.

Period Changes: $y = \sin(bx)$ and $y = \cos(bx)$

The coefficient $b$ inside the function changes how fast the function completes a cycle:

$$y = \sin(bx), \quad y = \cos(bx).$$

For sine and cosine:

• Period of $\sin x$ and $\cos x$ is $2\pi$.
• Period of $\sin(bx)$ and $\cos(bx)$ is
  $$\text{Period} = \dfrac{2\pi}{|b|}.$$

Interpretation:

• If $|b| > 1$, the graph completes its cycles faster → period is shorter → graph is horizontally compressed.
• If $0 < |b| < 1$, the graph completes its cycles more slowly → period is longer → graph is horizontally stretched.

Examples:

• $y = \sin(2x)$ has period $\dfrac{2\pi}{2} = \pi$.
• $y = \cos\left(\dfrac{x}{3}\right)$ has period $\dfrac{2\pi}{1/3} = 6\pi$.

Phase Shifts and Vertical Shifts

The most general basic forms for sine and cosine are:

$$y = a \sin(bx - c) + d,$$
$$y = a \cos(bx - c) + d.$$

Each parameter has a clear geometric meaning.

Vertical Shift $d$

The $+d$ at the end moves the graph up or down:

• Graph of $y = \sin x + 2$ is the graph of $\sin x$ shifted up by 2 units.
• Graph of $y = \cos x - 3$ is the graph of $\cos x$ shifted down by 3 units.

The line $y = d$ becomes the new “midline” around which the graph oscillates.

• Midline: $y = d$.
• Range: $[d - |a|, d + |a|]$.

Phase Shift $c$

The expression $(bx - c)$ inside the function shifts the graph left or right. To see the amount of horizontal shift, factor out $b$:

$$y = a\sin(bx - c) + d = a\sin\bigl(b(x - \tfrac{c}{b})\bigr) + d.$$

The phase shift is $\dfrac{c}{b}$:

• If $\dfrac{c}{b} > 0$: shift right by $\dfrac{c}{b}$.
• If $\dfrac{c}{b} < 0$: shift left by $|\dfrac{c}{b}|$.

Example:

• $y = \sin\left(x - \dfrac{\pi}{4}\right)$: $b = 1$, $c = \dfrac{\pi}{4}$, so shift right by $\dfrac{\pi}{4}$.
• $y = 3\cos\left(2x + \pi\right)$:
• Write inside as $2x + \pi = 2(x + \dfrac{\pi}{2})$.
• So it is $3\cos\bigl(2(x + \dfrac{\pi}{2})\bigr)$.
• Phase shift: left by $\dfrac{\pi}{2}$.
• Period: $\dfrac{2\pi}{2} = \pi$.
• Amplitude: $3$.
• Midline: $y = 0$ (no vertical shift).

Tangent Function

Definition, Domain, and Range

The tangent function is defined by:

$$\tan x = \dfrac{\sin x}{\cos x}.$$

Because division by zero is not allowed, $\tan x$ is undefined where $\cos x = 0$, i.e. at

$$x = \dfrac{\pi}{2} + k\pi,\quad k \in \mathbb{Z}.$$

Thus:

• Domain of $\tan x$:
  $$\mathbb{R} \setminus \left\{\dfrac{\pi}{2} + k\pi : k \in \mathbb{Z}\right\}.$$
• Range of $\tan x$:
  $$(-\infty, \infty).$$

As $x$ approaches the values where $\cos x = 0$, $\tan x$ grows without bound in the positive or negative direction. These are vertical asymptotes of the graph.

Periodicity and Symmetry

Tangent has a smaller period than sine and cosine:

• Fundamental period: $\pi$:
  $$\tan(x + \pi) = \tan x \quad\text{for all } x\text{ where both sides are defined}.$$

Tangent is an odd function:

• $\tan(-x) = -\tan x$.

So, like sine, its graph is symmetric about the origin.

Basic Graph of $y = \tan x$

To sketch one period (say from $-\dfrac{\pi}{2}$ to $\dfrac{\pi}{2}$):

• Vertical asymptotes at $x = -\dfrac{\pi}{2}$ and $x = \dfrac{\pi}{2}$.
• It passes through the origin: $(0, 0)$.
• Using special angles from the unit circle:
• $\tan \dfrac{\pi}{4} = 1$.
• $\tan \left(-\dfrac{\pi}{4}\right) = -1$.

The graph rises from $-\infty$ (approaching the left asymptote) to $+\infty$ (approaching the right asymptote) in a smooth S-shaped curve through $(0,0)$.

This pattern repeats every $\pi$ units.

Transformations of Tangent: $y = a \tan(bx)$

The function
$$y = a \tan(bx)$$
has:

• Vertical stretch factor $a$:
• For $a > 0$, the curve is steeper when $|a| > 1$, flatter when $0<|a|<1$.
• If $a < 0$, the graph is reflected across the $x$-axis.
• Period:
• Basic $\tan x$ period is $\pi$.
• For $\tan(bx)$, period is
  $$\text{Period} = \dfrac{\pi}{|b|}.$$
• Vertical asymptotes occur where $bx = \dfrac{\pi}{2} + k\pi$, so
  $$x = \dfrac{1}{b}\Bigl(\dfrac{\pi}{2} + k\pi\Bigr).$$

Including shifts, the general form is:

$$y = a \tan(bx - c) + d.$$

• The phase shift is $\dfrac{c}{b}$ (similar to sine and cosine).
• The vertical shift $d$ moves the central point of the S-curve up or down, and moves the center line to $y = d$.

Summary of Key Forms and Properties

For $y = a\sin(bx - c) + d$ or $y = a\cos(bx - c) + d$:

• Amplitude: $|a|$.
• Period: $\dfrac{2\pi}{|b|}$.
• Phase shift: $\dfrac{c}{b}$ (right if positive, left if negative).
• Vertical shift: $d$.
• Midline: $y = d$.
• Range: $[d - |a|, d + |a|]$.

For $y = a\tan(bx - c) + d$:

• No finite amplitude (range is all real numbers).
• Period: $\dfrac{\pi}{|b|}$.
• Phase shift: $\dfrac{c}{b}$.
• Vertical shift: $d$.
• Midline: $y = d$.
• Vertical asymptotes at
  $$bx - c = \dfrac{\pi}{2} + k\pi \quad\Rightarrow\quad x = \dfrac{c}{b} + \dfrac{1}{b}\Bigl(\dfrac{\pi}{2} + k\pi\Bigr).$$

In later chapters (on identities and further applications) you will use these function properties together with algebraic identities to solve equations, model real-world periodic phenomena, and analyze more complex trigonometric expressions.