Chapter Contents

• Graphs of the Trigonometric Functions
• 1. Graphs of y = a sin x and y = a cos x
• 2. Graphs of y = a sin bx and y = a cos bx
• Frequency of Music Notes
• 3. Graphs of y = a sin(bx + c) and y = a cos(bx + c)
• 4. Graphs of tan, cot, sec and csc
• 5. Applications of Trigonometric Graphs
• 6. Composite Trigonometric Graphs
• Biorhythm Graphs
• 7. Lissajous Figures
• Graphs of the Trigonometric Functions Problem Solver

• Comments, Questions?

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5. Applications of Trigonometric Graphs

by M. Bourne

Simple Harmonic Motion

Any object moving with constant angular velocity or moving up and down with a regular motion can be described in terms of SIMPLE HARMONIC MOTION.

The displacement, d, of an object moving with SHM, is given by:

d = R sin ωt

where R is the radius of the rotating object and ω is the angular velocity of the object.

For an animation of this concept, go back to: sin animation.

NOTE: We may need to use one the following, depending on the situation:

d = R cos ωt

d = R sin (ωt + α)

d = R cos (ωt + α)

Example 1

A point on a cam is 8.30 cm from the centre of rotation. Sketch 2 cycles of d as a function of t, given that d = 0 cm when t = 0 s and ω = 3.20 rad/s.

Answer

Example 2

The voltage of an alternating current circuit is given by

e = E cos(ωt + α).

Sketch 2 cycles of the voltage as a function of time if

E = 80 V, ω = 377 rad/s and α = π/2.

Answer

Example 3

The signal received by a radio is given by

e = 0.014 cos(2πft),

where e is in volts and f is in Hz.

Draw 2 cycles of e for f = 950 kHz.

Answer

Angular Velocity

Another important result in this section is:

The angular velocity ω (in radians per second) of a rotating object, is given by:

ω = 2πf

where f is the frequency of the motion, in cycles per second.

Exercises

1. A satellite is orbiting the earth so that its displacement D north of the equator is given by

D = A sin(ωt + α).

Sketch 2 cycles of D as a function of t if

A = 500 km, ω = 3.60 rad/hr and α = 0.

Answer

2. Using e = E cos(ωt + α), sketch 2 cycles of the voltage as a function of time if

E = 170 V, ω = 120π rad/s and `α = π/3`.

Answer