Graphs

Understanding Graphs of Trigonometric Functions

In this chapter we focus on how the basic trigonometric functions look when drawn on the coordinate plane, and what their main graphical features are. The functions of interest are $y = \sin x$, $y = \cos x$, and $y = \tan x$. Ideas such as “sine”, “cosine”, “tangent”, “angle”, and “radian measure” are assumed from earlier chapters; here we concentrate on their graphs.

Throughout this chapter, unless otherwise stated, angles are measured in radians and $x$ is the angle input to the trig function.

General Features of Trig Graphs

All trigonometric functions share some common graphical behaviors:

• They repeat regularly (they are periodic).
• They have characteristic shapes that repeat every period.
• Many have symmetries: some are symmetric about the $y$-axis, others about the origin.
• They often have a maximum and minimum “height” (amplitude), or they may grow arbitrarily large in magnitude between vertical asymptotes.

We now look at each basic function in turn.

Graph of $y = \sin x$

The sine function takes an angle $x$ and outputs the $y$–coordinate of the corresponding point on the unit circle. Graphically:

• The input axis (horizontal) is the angle $x$.
• The output axis (vertical) is $\sin x$.

Key Points for $y = \sin x$

Over one full “cycle” from $x = 0$ to $x = 2\pi$, some especially useful points are:

• $x = 0$: $\sin 0 = 0$
• $x = \dfrac{\pi}{2}$: $\sin \dfrac{\pi}{2} = 1$ (maximum in this cycle)
• $x = \pi$: $\sin \pi = 0$
• $x = \dfrac{3\pi}{2}$: $\sin \dfrac{3\pi}{2} = -1$ (minimum in this cycle)
• $x = 2\pi$: $\sin 2\pi = 0$

Plotting these points and smoothly connecting them gives the familiar “wave” shape.

Other notable points include

• $x = \dfrac{\pi}{6}$: $\sin \dfrac{\pi}{6} = \dfrac{1}{2}$,
• $x = \dfrac{\pi}{4}$: $\sin \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$,
• $x = \dfrac{\pi}{3}$: $\sin \dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2}$,
  and their reflections in later quadrants.

Period, Amplitude, and Range

For $y = \sin x$:

• Period: the smallest positive number $P$ such that
  $$\sin(x + P) = \sin x \quad \text{for all } x$$
  is
  $$P = 2\pi.$$
• Amplitude: the maximum distance from the midline (here, the $x$-axis) to the peak of the graph. For $y = \sin x$ the amplitude is
  $$\text{Amplitude} = 1.$$
• Range: the set of all possible output values:
  $$-1 \le \sin x \le 1.$$

Zeros and Symmetry

• Zeros (x-intercepts): $\sin x = 0$ whenever $x$ is an integer multiple of $\pi$:
  $$x = n\pi \quad \text{for any integer } n.$$
• Symmetry: $\sin x$ is an odd function:
  $$\sin(-x) = -\sin x.$$
  Graphically, this means the graph of $y = \sin x$ is symmetric with respect to the origin.

Sketching One Period of $y = \sin x$

To sketch $y = \sin x$ on $[0, 2\pi]$:

1. Mark the zeros at $0,\ \pi,\ 2\pi$.
2. Mark the maximum at $\left(\dfrac{\pi}{2}, 1\right)$.
3. Mark the minimum at $\left(\dfrac{3\pi}{2}, -1\right)$.
4. Draw a smooth, continuous curve passing through these points, making sure the shape is smooth (no corners) and wave-like.

Then extend the pattern left and right using periodicity with period $2\pi$.

Graph of $y = \cos x$

The cosine function takes an angle $x$ and outputs the $x$–coordinate of the corresponding point on the unit circle.

Key Points for $y = \cos x$

Over one cycle from $x = 0$ to $x = 2\pi$:

• $x = 0$: $\cos 0 = 1$ (maximum in this cycle)
• $x = \dfrac{\pi}{2}$: $\cos \dfrac{\pi}{2} = 0$
• $x = \pi$: $\cos \pi = -1$ (minimum in this cycle)
• $x = \dfrac{3\pi}{2}$: $\cos \dfrac{3\pi}{2} = 0$
• $x = 2\pi$: $\cos 2\pi = 1$

As with sine, we can add intermediate unit-circle values:

• $x = \dfrac{\pi}{3}$: $\cos \dfrac{\pi}{3} = \dfrac{1}{2}$,
• $x = \dfrac{\pi}{4}$: $\cos \dfrac{\pi}{4} = \dfrac{\sqrt{2}}{2}$,
• $x = \dfrac{\pi}{6}$: $\cos \dfrac{\pi}{6} = \dfrac{\sqrt{3}}{2}$,
  and the corresponding values in other quadrants.

Period, Amplitude, and Range

For $y = \cos x$:

• Period:
  $$P = 2\pi.$$
• Amplitude:
  $$\text{Amplitude} = 1.$$
• Range:
  $$-1 \le \cos x \le 1.$$

Zeros and Symmetry

• Zeros (x-intercepts): $\cos x = 0$ when
  $$x = \dfrac{\pi}{2} + n\pi \quad \text{for any integer } n.$$
• Symmetry: $\cos x$ is an even function:
  $$\cos(-x) = \cos x.$$
  The graph is symmetric with respect to the $y$-axis.

Relation Between Sine and Cosine Graphs

The graphs of sine and cosine are the same shape, shifted horizontally:

• Using the identity
  $$\sin\!\left(x + \dfrac{\pi}{2}\right) = \cos x,$$
  we can see $y = \cos x$ is the graph of $y = \sin x$ shifted to the left by $\dfrac{\pi}{2}$ units.

Recognizing this phase shift helps you move between the sine and cosine graphs quickly.

Sketching One Period of $y = \cos x$

To sketch $y = \cos x$ on $[0, 2\pi]$:

1. Start at $\left(0, 1\right)$.
2. Pass through $\left(\dfrac{\pi}{2}, 0\right)$.
3. Reach the minimum at $\left(\pi, -1\right)$.
4. Pass again through $\left(\dfrac{3\pi}{2}, 0\right)$.
5. Return to $\left(2\pi, 1\right)$.

Connect these smoothly in a wave-like curve and extend by repeating the pattern every $2\pi$.

Graph of $y = \tan x$

The tangent function is defined (for angles where it is defined) by
$$\tan x = \dfrac{\sin x}{\cos x}.$$
Its graph looks quite different from sine and cosine:

• It is not bounded (it takes arbitrarily large positive and negative values).
• It has vertical asymptotes where $\cos x = 0$.
• Its basic pattern repeats more frequently.

Domain and Vertical Asymptotes

Tangent is undefined where $\cos x = 0$, i.e. at
$$x = \dfrac{\pi}{2} + n\pi, \quad \text{for any integer } n.$$

At these $x$-values, the graph has vertical asymptotes: the curve approaches these lines but never touches or crosses them.

So:

• Domain of $y = \tan x$: all real numbers except $x = \dfrac{\pi}{2} + n\pi$.

Key Points for $y = \tan x$

Tangent has period $\pi$ (rather than $2\pi$). On the interval $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$:

• $x = 0$: $\tan 0 = 0$
• As $x \to \dfrac{\pi}{2}^-$ (approach from the left), $\tan x \to +\infty$.
• As $x \to -\dfrac{\pi}{2}^+$ (approach from the right), $\tan x \to -\infty$.

Useful intermediate angles:

• $x = \dfrac{\pi}{4}$: $\tan \dfrac{\pi}{4} = 1$.
• $x = -\dfrac{\pi}{4}$: $\tan\!\left(-\dfrac{\pi}{4}\right) = -1$.

In each interval between asymptotes, the graph is a smooth, increasing curve passing from $-\infty$ to $+\infty$.

Period and Range

For $y = \tan x$:

• Period: the smallest positive $P$ such that $\tan(x + P) = \tan x$ is
  $$P = \pi.$$
• Range: tangent can take any real value:
  $$\text{Range of } \tan x = (-\infty, \infty).$$

Zeros and Symmetry

• Zeros (x-intercepts): $\tan x = 0$ whenever $\sin x = 0$ and $\cos x \ne 0$, i.e.
  $$x = n\pi \quad \text{for any integer } n.$$
• Symmetry: $\tan x$ is an odd function:
  $$\tan(-x) = -\tan x.$$
  The graph is symmetric with respect to the origin.

Sketching One Period of $y = \tan x$

To sketch $y = \tan x$ on $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$:

1. Draw vertical dashed lines (asymptotes) at $x = -\dfrac{\pi}{2}$ and $x = \dfrac{\pi}{2}$.
2. Plot the point $\left(0, 0\right)$.
3. Plot points $\left(\dfrac{\pi}{4}, 1\right)$ and $\left(-\dfrac{\pi}{4}, -1\right)$.
4. Draw a smooth curve that
• passes through these points,
• approaches $-\infty$ as $x$ approaches $-\dfrac{\pi}{2}$ from the right,
• and approaches $+\infty$ as $x$ approaches $\dfrac{\pi}{2}$ from the left.
5. Repeat this pattern, shifted left and right by $\pi$.

Transformations of Trigonometric Graphs

More general trigonometric functions often appear in the form
$$y = A \sin(Bx + C) + D,$$
or similarly for cosine and tangent. Details of transformation rules are treated elsewhere, but you should recognize, in terms of graphs:

• $A$ affects the amplitude (vertical stretch/flip for sine and cosine).
• $B$ affects the period: for sine and cosine, the period becomes
  $$\text{new period} = \dfrac{2\pi}{|B|};$$
  for tangent, it becomes
  $$\text{new period} = \dfrac{\pi}{|B|}.$$
• $C$ shifts the graph horizontally (phase shift).
• $D$ shifts the graph vertically (moves the midline up or down).

Even before mastering all transformation formulas, you should be able to:

• Recognize the base shape (sine-like, cosine-like, or tangent-like).
• Identify the period length on the $x$-axis.
• Identify the “height” of the peaks and troughs (for sine and cosine) and the new positions of asymptotes (for tangent).

Comparing the Three Basic Graphs

To keep the key ideas in one place, here is a side-by-side comparison:

• $y = \sin x$
• Shape: smooth wave crossing the origin.
• Period: $2\pi$.
• Amplitude: $1$.
• Range: $[-1, 1]$.
• Zeros: $x = n\pi$.
• Symmetry: odd, origin symmetry.
• $y = \cos x$
• Shape: same wave as sine, shifted left by $\dfrac{\pi}{2}$.
• Period: $2\pi$.
• Amplitude: $1$.
• Range: $[-1, 1]$.
• Zeros: $x = \dfrac{\pi}{2} + n\pi$.
• Symmetry: even, $y$-axis symmetry.
• $y = \tan x$
• Shape: repeating S-shaped curves between vertical asymptotes.
• Period: $\pi$.
• Amplitude: no fixed amplitude (unbounded).
• Range: $(-\infty, \infty)$.
• Zeros: $x = n\pi$.
• Vertical asymptotes: $x = \dfrac{\pi}{2} + n\pi$.
• Symmetry: odd, origin symmetry.

A firm grasp of these graphical features will prepare you for exploring more advanced trigonometric functions, transformations, and identities, and for applying trig graphs to real-world periodic phenomena.