5.3 Trigonometric graphs and transformations

Sine and cosine model smooth periodic change, while tangent models ratios with regularly spaced vertical asymptotes.

5.3.1 Parent graphs and domains

Sine and cosine have domain \(\mathbb R\), range \([-1,1]\), and period \(2\pi\). Tangent has period \(\pi\), zeros at \(k\pi\), and is undefined at \(\pi/2+k\pi\). Cosecant, secant, and cotangent inherit exclusions from their reciprocal definitions.

Worked Example: Reading the sine graph

Over \([0,2\pi]\), sine passes through 0 at \(0,\pi,2\pi\), reaches 1 at \(\pi/2\), and reaches \(-1\) at \(3\pi/2\).

Try It. List the corresponding five key values for cosine.
Check Your Work \(1,0,-1,0,1\) at \(0,\pi/2,\pi,3\pi/2,2\pi\).

Worked Example: Finding tangent exclusions

\(\tan x=\sin x/\cos x\) is undefined wherever \(\cos x=0\), so \(x=\pi/2+k\pi\) is excluded.

Try It. Where is \(\sec x\) undefined?
Check Your Work At \(x=\pi/2+k\pi\).

5.3.2 Amplitude, period, and phase shift

For \(y=a\sin(kx-b)+d\) or \(a\cos(kx-b)+d\), amplitude is \(|a|\), period is \(2\pi/|k|\), phase shift is \(b/k\), and midline is \(y=d\). Apply horizontal changes before vertical changes.

Worked Example: Analyzing a transformed sine curve

For \(y=3\sin[2(x-\pi/4)]\), amplitude is 3, period is \(\pi\), phase shift is \(\pi/4\) right, and midline is 0.

Try It. Analyze \(y=2\cos(4x)+1\).
Check Your Work Amplitude 2, period \(\pi/2\), no phase shift, midline \(y=1\).

Worked Example: Modelling periodic tank depth

A depth varying between 2 m and 6 m has midline 4 and amplitude 2. If its period is 12 h and it begins at a maximum,

\[d(t)=4+2\cos(\pi t/6).\]

Try It. Write a cosine model with minimum 1, maximum 5, period 8, beginning at a maximum.
Check Your Work \(y=3+2\cos(\pi t/4)\).
Applied Problem: Modelling daily reservoir depth. Reservoir depth ranges from 7.2 m to 9.8 m with a 24 h period and is highest at midnight. Write a cosine model and find the predicted depth 6 h later.
Check Your Work Midline \(8.5\), amplitude \(1.3\), and angular frequency \(2\pi/24=\pi/12\). Thus \(d(t)=8.5+1.3\cos(\pi t/12)\). At 6 h, \(d=8.5\) m.

5.3.3 Practice Problems

  1. State the period and range of sine.
    Check Your Work Period \(2\pi\); range \([-1,1]\).
  2. State the period of tangent.
    Check Your Work \(\pi\).
  3. Find the amplitude and period of \(4\sin(3x)\).
    Check Your Work Amplitude 4; period \(2\pi/3\).
  4. Find the midline of \(-2\cos x+5\).
    Check Your Work \(y=5\).
  5. Find the phase shift of \(\sin[2(x-1)]\).
    Check Your Work 1 unit right.
  6. Water level ranges from 3 m to 11 m. Find amplitude and midline.
    Check Your Work Amplitude 4 m; midline 7 m.
  7. A backwash pressure cycle has period 10 min. Find its angular frequency \(k\).
    Check Your Work \(k=2\pi/10=\pi/5\) rad/min.
  8. A depth model is \(d(t)=5+1.5\sin(\pi t/6)\). Find its maximum, minimum, and period.
    Check Your Work Maximum 6.5 m, minimum 3.5 m, period 12 h.