What is the difference between sine and cosine graphs?

The sine and cosine graphs have the same shape, amplitude (1), and period (2π). The only difference is a horizontal shift: cosine leads sine by π/2 radians. In other words, cos x = sin(x + π/2). Sine passes through the origin, while cosine starts at its maximum value of 1.

Why does the tangent graph have asymptotes?

Tangent is defined as sin x / cos x. Whenever cos x = 0, the fraction is undefined, so the function has no value there. On the graph, these undefined points appear as vertical asymptotes — lines the curve approaches but never touches. This happens at x = π/2 + nπ for every integer n.

How do you find the period of a trig graph from its equation?

For sine and cosine in the form y = A sin(Bx) or y = A cos(Bx), divide 2π by |B|. For tangent in the form y = A tan(Bx), divide π by |B|. A larger B compresses the graph horizontally, producing a shorter period.

Graphs of Sine, Cosine, and Tangent — Definition, Formula & Examples

Graphs of sine, cosine, and tangent are the visual representations of the three primary trigonometric functions plotted on the coordinate plane. Each graph has a distinctive, repeating shape: sine and cosine produce smooth waves, while tangent produces a series of curves separated by vertical asymptotes.

The graph of $y = \sin x$ is a continuous, periodic wave oscillating between $-1$ and $1$ with period $2\pi$ . The graph of $y = \cos x$ is identical in shape but shifted $\frac{\pi}{2}$ units to the left. The graph of $y = \tan x$ has period $\pi$ , no maximum or minimum values, and vertical asymptotes wherever $\cos x = 0$ , namely at $x = \frac{\pi}{2} + n\pi$ for every integer $n$ .

Key Formula

y = A\sin(Bx - C) + D \qquad y = A\cos(Bx - C) + D \qquad y = A\tan(Bx - C) + D

Where:

$A$ = Amplitude (for sine and cosine); vertical stretch factor (for tangent)
$B$ = Affects the period: period = 2π/|B| for sin/cos, π/|B| for tan
$C$ = Horizontal (phase) shift: the graph shifts right by C/B
$D$ = Vertical shift: moves the midline up or down by D units

How It Works

To sketch any of these graphs, identify four key features: amplitude, period, phase shift, and vertical shift. For

y = A\sin(Bx - C) + D

y = A\cos(Bx - C) + D

, the amplitude is

|A|

, the period is

\frac{2\pi}{|B|}

, the phase shift is

\frac{C}{B}

, and the vertical shift is

D

. Sine starts at the midline and rises; cosine starts at its maximum. For tangent,

y = A\tan(Bx - C) + D

, there is no amplitude because tangent has no bounded range; instead, the period is

\frac{\pi}{|B|}

and asymptotes appear at the endpoints of each period. Plot one full cycle by marking key points — zeros, peaks, troughs for sine/cosine, or the center zero and two asymptotes for tangent — then repeat the pattern in both directions.

Worked Example

Problem: Sketch one full cycle of y = 2 sin(3x) and state the amplitude and period.

Step 1: Identify A and B from the equation. Here A = 2 and B = 3.

y = 2\sin(3x)

Step 2: Find the amplitude. The amplitude is |A| = 2, so the graph oscillates between −2 and 2.

\text{Amplitude} = |A| = 2

Step 3: Find the period. Divide 2π by |B|.

\text{Period} = \frac{2\pi}{|3|} = \frac{2\pi}{3}

Step 4: Mark key x-values for one cycle by dividing the period into four equal parts: 0, π/6, π/3, π/2, and 2π/3.

0,\;\frac{\pi}{6},\;\frac{\pi}{3},\;\frac{\pi}{2},\;\frac{2\pi}{3}

Step 5: Plot the key points: (0, 0), (π/6, 2), (π/3, 0), (π/2, −2), (2π/3, 0). Connect them with a smooth wave.

\left(0,0\right),\;\left(\tfrac{\pi}{6},2\right),\;\left(\tfrac{\pi}{3},0\right),\;\left(\tfrac{\pi}{2},-2\right),\;\left(\tfrac{2\pi}{3},0\right)

Answer: The graph of y = 2 sin(3x) is a sine wave with amplitude 2 and period 2π/3.

Another Example

This example covers tangent rather than sine, showing how to handle vertical asymptotes and the different period formula.

Problem: Sketch one full cycle of y = tan(2x) and identify the period and asymptotes.

Step 1: Identify B from the equation. Here B = 2.

y = \tan(2x)

Step 2: Find the period of tangent. Divide π by |B|.

\text{Period} = \frac{\pi}{|2|} = \frac{\pi}{2}

Step 3: Locate the vertical asymptotes. For standard tangent, asymptotes occur at the edges of each period. Center one cycle at x = 0, so asymptotes are at x = −π/4 and x = π/4.

x = -\frac{\pi}{4},\quad x = \frac{\pi}{4}

Step 4: Plot three key points: the center zero and the halfway points. At x = 0, y = 0. At x = π/8, y = tan(π/4) = 1. At x = −π/8, y = −1.

\left(-\tfrac{\pi}{8},-1\right),\;\left(0,0\right),\;\left(\tfrac{\pi}{8},1\right)

Step 5: Draw the S-shaped curve through these points, approaching the asymptotes at both ends. Repeat every π/2 units.

Answer: The graph of y = tan(2x) has period π/2 with vertical asymptotes at x = π/4 + nπ/2 for every integer n.

Visualization

Why It Matters

Graphing sine, cosine, and tangent is a core skill in Precalculus and AP Calculus, where you analyze derivatives and integrals of trig functions visually. Engineers and physicists model sound waves, alternating current, and pendulum motion using these graphs. Understanding the shape and features of each curve also prepares you for Fourier analysis, which decomposes complex signals into sums of sine and cosine waves.

Common Mistakes

Mistake: Using the sine/cosine period formula (2π/|B|) for tangent.

Correction: Tangent's period is π/|B|, half as long as sine's. Always check which function you are graphing before computing the period.

Mistake: Confusing amplitude with the vertical shift.

Correction: Amplitude (|A|) measures how far the wave stretches above and below the midline. The vertical shift (D) moves the entire midline up or down. They are independent parameters.

Mistake: Placing tangent asymptotes at multiples of π instead of at π/2 + nπ.

Correction: For y = tan x, cos x = 0 at x = π/2 + nπ, not at x = nπ. At x = nπ, tangent actually equals zero.

Check Your Understanding

What are the amplitude and period of y = 3 cos(4x)?

Hint: Use amplitude = |A| and period = 2π / |B|.

Answer: Amplitude = 3, Period = 2π/4 = π/2.

Where are the asymptotes of y = tan(x/2) in the interval [−2π, 2π]?

Hint: Set the argument x/2 equal to π/2 + nπ and solve for x.

Answer: At x = −π, x = π, and x = 3π (within the interval). Asymptotes occur at x = π + 2nπ for every integer n.

True or false: The graph of y = −sin x is the same as y = sin x reflected over the x-axis.

Hint: Think about what a negative A does to each output value.

Answer: True. Multiplying by −1 flips every y-value, reflecting the wave over the x-axis.

Related Terms

Amplitude — Height of a sine or cosine wave from the midline
Period of a Periodic Function — Length of one complete cycle of a trig graph
Phase Shift — Horizontal translation of a trig graph
Periodic Function — Parent concept: functions that repeat at regular intervals
Frequency of a Periodic Function — Number of cycles per unit, the reciprocal of the period
Frequency of Periodic Motion — Physical application of frequency in wave and motion problems
Periodic Motion — Real-world motion modeled by trig graphs
Period of Periodic Motion — Time for one complete oscillation in a physical system
Model — Trig graphs are used to model cyclical data
RPM — Rotational speed often converted to angular frequency for trig graphs