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Precalculus Unit 7: Graphs of trig functions

Sine and cosine graphs are waves. Four numbers control them: amplitude, period, phase shift and vertical shift.

Covers: Sinusoidal amplitude/period/phase, tangent, sinusoidal modeling

Study guide

The graphs of sine and cosine are sinusoids: smooth waves that repeat every 2π. A general sinusoid is f(θ) = a sin(b(θ + c)) + d. The amplitude |a| is half the distance from maximum to minimum. The period is 2π/|b|, the length of one full cycle. The phase shift c moves the graph horizontally (left c if written θ + c). The midline y = d is the average of the maximum and minimum values.

Cosine is just sine shifted left by π/2, so either can model the same wave. To build a model from data or a graph, find the max and min to get the midline and amplitude, measure the distance between peaks for the period, and locate a starting point (a max for cosine, a midline crossing going up for sine) for the phase shift.

The tangent function, tan θ = sin θ/cos θ, has period π, passes through the origin, and has vertical asymptotes wherever cos θ = 0, at θ = π/2 + πk. Between asymptotes it increases through all real numbers.

Sinusoidal modeling applies these ideas to periodic phenomena such as tides, daylight hours, Ferris wheels and sound waves. Interpret each parameter in context: the midline is the average level, the amplitude is the swing above and below it, and the period is the time for one cycle.

A sinusoid changes fastest at its midline crossings and has a rate of change of zero at its maximum and minimum points. It is concave down where the graph is above the midline and concave up where it is below, so the rate of change is decreasing on the way from an upward midline crossing to the next downward one. Frequency is the number of cycles per unit of input, 1/period = |b|/(2π). A larger |b| means more cycles in the same interval.

Key ideas

• y = a·sin(b(x − c)) + d: |a| is the amplitude, 2π/|b| is the period, c is the phase shift, d is the midline.
• Sine starts at the midline going up; cosine starts at a maximum.
• tan x has period π and vertical asymptotes where cos x = 0.
• Sinusoidal models fit repeating data like tides, temperatures and daylight hours.

Formulas and rules

Common mistakes

• Taking b as the period instead of 2π/b.
• Forgetting to factor b out before reading the phase shift.

Words to know

Period of sin(bx)

2π/|b|

Amplitude

|a| in y = a sin x or y = a cos x: half the distance from max to min.

Midline

y = (max + min)/2

Period of tan x

π

Phase shift

For sin(b(x − c)), a shift of c units to the right.

Sinusoid

A sine- or cosine-shaped wave.

Period

The length of one full cycle.

Frequency

Cycles per unit: 1/period.

Tangent asymptotes

At θ = π/2 + πk, where cos θ = 0.

Sinusoidal model

f(t) = a sin(b(t − c)) + d for periodic data like tides.

Cosine vs. sine

cos θ = sin(θ + π/2): cosine is sine shifted left π/2.

Sample practice questions

23 more questions like these, from level 1 to 5, are in the practice set.