Trigonometric Graphs: Sine, Cosine & Tangent

Why Study Trigonometric Graphs?

The graphs of $\sin x$, $\cos x$ and $\tan x$ are among the most important functions in mathematics and physics. They model everything from sound waves and alternating current to tidal patterns and planetary orbits. Understanding how to read and manipulate these graphs is essential at A-Level, AP and IB level.

Learning Goals: By the end of this guide, you should be able to:

1. Sketch the graphs of $\sin x$, $\cos x$ and $\tan x$ from memory.
2. Identify amplitude, period, phase shift and vertical translation from an equation.
3. Write the equation of a trigonometric function from its graph.
4. Solve graphical problems involving trigonometric transformations.

The Three Parent Graphs

$y = \sin x$

• Shape: Smooth wave starting at the origin
• Range: $[-1, 1]$
• Period: $2\pi$ (or $360°$)
• Key points: $(0, 0)$, $(\frac{\pi}{2}, 1)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -1)$, $(2\pi, 0)$

$y = \cos x$

• Shape: Smooth wave starting at maximum
• Range: $[-1, 1]$
• Period: $2\pi$ (or $360°$)
• Key points: $(0, 1)$, $(\frac{\pi}{2}, 0)$, $(\pi, -1)$, $(\frac{3\pi}{2}, 0)$, $(2\pi, 1)$
• Relationship: $\cos x = \sin\left(x + \frac{\pi}{2}\right)$ — cosine is sine shifted left by $\frac{\pi}{2}$.

$y = \tan x$

• Shape: Repeating S-curves between vertical asymptotes
• Range: $(-\infty, +\infty)$
• Period: $\pi$ (or $180°$)
• Asymptotes at $x = \frac{\pi}{2} + n\pi$ where $n$ is any integer

The General Equation

All three trigonometric graphs follow the general form:

(Replace $\sin$ with $\cos$ or $\tan$ as appropriate.)

Worked Examples

Example 1: Reading Parameters from an Equation

Question: For $y = 3\sin(2x - \pi) + 1$, find the amplitude, period, phase shift and midline.

Step 1: Rewrite in standard form: $y = 3\sin\left(2\left(x - \frac{\pi}{2}\right)\right) + 1$

Step 2: Read off the parameters:

• Amplitude: $|A| = 3$
• Period: $\frac{2\pi}{|B|} = \frac{2\pi}{2} = \pi$
• Phase shift: $C = \frac{\pi}{2}$ (shift right)
• Midline: $y = 1$

Example 2: Writing the Equation from a Graph

Question: A cosine curve has maximum 5, minimum 1, period $4\pi$, and starts at its maximum when $x = \pi$.

Step 1: Amplitude $= \frac{5 - 1}{2} = 2$, midline $D = \frac{5 + 1}{2} = 3$

Step 2: Period $= 4\pi$, so $B = \frac{2\pi}{4\pi} = \frac{1}{2}$

Step 3: Maximum at $x = \pi$, so phase shift $C = \pi$

Answer: $y = 2\cos\left(\frac{1}{2}(x - \pi)\right) + 3$

Sine vs Cosine: The Phase Relationship

The key identity connecting sine and cosine is:

This means any cosine graph can be rewritten as a sine graph (and vice versa) by adjusting the phase shift. Examiners may accept either form.

Common Mistakes

1. Forgetting to factor out $B$ before reading the phase shift — in $y = \sin(2x - \pi)$, the phase shift is NOT $\pi$. You must write $\sin(2(x - \frac{\pi}{2}))$ to see the true shift is $\frac{\pi}{2}$.
2. Confusing period with frequency — period is the length of one cycle; frequency ($B$) is the number of cycles in $2\pi$.
3. Drawing tangent without asymptotes — always mark asymptotes first, then sketch the curve between them.

Exam Tips (A-Level / AP / IB)

• A quick way to find the period from a graph: measure the horizontal distance between two consecutive peaks (or troughs, or any two corresponding points).
• When sketching, always mark the five key points of one cycle first (start, quarter, half, three-quarter, end), then connect smoothly.
• For inverse trig questions, remember the restricted domains: $\sin^{-1}$ returns $[-\frac{\pi}{2}, \frac{\pi}{2}]$, $\cos^{-1}$ returns $[0, \pi]$, $\tan^{-1}$ returns $(-\frac{\pi}{2}, \frac{\pi}{2})$.

Frequently Asked Questions

Why is the period of tangent $\pi$ instead of $2\pi$?

Because $\tan x = \frac{\sin x}{\cos x}$, and after shifting by $\pi$, both $\sin$ and $\cos$ have been negated — their ratio remains unchanged. So $\tan(x + \pi) = \tan x$.

How do I solve trigonometric equations graphically?

Draw the trigonometric curve and a horizontal line at the target value. The x-coordinates of intersection points are your solutions. Use periodicity to find all solutions in the required interval.

Related Topics

• Function Transformations — The general rules behind translations, stretches and reflections.
• Sequences & Series — Fourier series connect trigonometric functions with infinite sums.
• Complex Numbers — Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ unifies trigonometry and complex algebra.