Math 117 Lecture 10 notes page 1
REVIEW:
TRIGONOMETRY, as the word implies, is concerned with the measurement of the parts of a triangle.
Plane trigonometry is restricted to right triangles lying in planes. Spherical or circular trigonometry
deals with certain triangles that lie on circles.
Consider a right triangle, and choose one angle as the point of reference.
hypotenuse
point of reference
θ
opposite
adjacent
Ratios of the right triangle’s three sides are used to define the six trig functions:
sin θ =
!
csc θ =
opp
cos θ =
hyp
hyp
adj
tan θ =
hyp
hyp
sec θ =
!
adj
opp
opp
adj
adj
cot θ =
!
opp
Two special right triangles occur frequently in trigonometry. The ratios of their sides should be
learned.
!
!
!
30°
√2
1
2
√3
1
45°
Note that trig functions seem to go in cycles:
sin 0° = 0
sin 90° = 1
sin 30° = 1/2
sin 120° = √3/2
sin 45° = √2/2
sin 135° = √2/2
sin 60° = √3/2
sin 150° = 1/2
60°
1
sin
sin
sin
sin
180° = 0
210° = –1/2
225° = –√2/2
240° = –√3/2
sin
sin
sin
sin
270° = –1
300° = –√3/2
315° = –√2/2
330° = –1/2
sin 360° = 0
The values (decimal approx.) of trig functions can also be found using a calculator. (mode: degree)
One radian is the equivalent measure of an angle in standard position
whose terminal side intercepts an arc of length r (radius).
r
1 radian
Because the circumference of a circle is 2πr, there are 2π radians
in a full circle. Degree measure and radian measure are therefore
related by the equation: 360° = 2π radians, or 180° = π radians.
You can use these equations to convert degrees to radians and
radians to degrees.
For example: 30° = 30°• π rad/180° = π/6 rads
r
Math 117 Lecture 10 notes page 2
To find EXACT trig values means writing the answer with √ and not decimal approximations. Note: Use
the ratios of 30-60-90 or 45-45-90 triangles to get exact values.
For example: sin 60° = √3/2
or, tan π/4 = 1/√2
A calculator will give decimal approximations of trig values. Round to 4 decimal places.
For example: sin 75° = 0.9659 (degree mode)
or, tan 1.12 rad = 2.0660 (radian mode)
NEW TOPIC:
Taking a unit circle (radius = 1), we can graph a sine function, called a sinusoidal curve, on the x-y axes.
y
y
x
The domain of the sinusoidal function is all real numbers.
The range of the sinusoidal function is –1 ≤ y ≤ 1.
Each function is periodic, meaning it has a repeating pattern.
The shortest repeating portion is called a cycle. The horizontal length of a cycle is called the period.
The amplitude is 1/2 (Max – min).
The general sinusoidal function can be written:
y = a sin b(x – h) + k
or
y = a cos b(x – h) + k
where a is the amplitude, k is a vertical shift, h is a horizontal shift or phase shift, and the period is
2"
360°
depending whether the angles are measured in degrees or radians. (note: most
b
b
application’s are in radians)
!
or
The basic sinusoidal wave, y = sin x, has no vertical shift, no horizontal (phase) shift, amplitude = 1, and
!
period = 2π.
The difference between the graph of a cosine wave and graph of a sine wave is a phase shift of π/2.
The cosine wave “starts” at a max (or min) and the sine wave “starts” in the middle.
Graph the sinusoidal function y = 2 cos 2(x – 0) + 0 or y = 2 cos 2x
Math 117 Lecture 10 notes page 3
Strategy:
1.
Sketch in the vertical shift (sinusoidal axis)
2.
Sketch in the Max and min using the amplitude
3.
Calculate the phase shift and mark a starting point
4.
Calculate the period and mark an ending point of the cycle.
5.
Mark other key points in the cycle, then draw in the curve.
Try these:
Graph the sinusoidal function
y = 1.5 cos 0.5(x) + 1
Graph the sinusoidal function
y = 3 cos (x – π)
Graph the sinusoidal function
y = 2 cos 2(x – π/2) + 1
Write an equation for this sinusoidal curve.
Math 117 Lecture 10 notes page 4
Application #1:
The pressure P (in millimeters of mercury) against the walls of the blood vessels of a patient is given
by
P = 100 – 20 cos (8π/3) t
where t is the time (in seconds). If one cycle is equivalent to one heartbeat, what is the person’s pulse
rate in heartbeats per minute?
Application #2:
The motion of a simple spring can be modeled by y = A cos k t where y is the spring’s vertical
displacement (in feet) relative to its position at rest, A is the initial displacement (in feet), k is a
constant that measures the elasticity of the spring, and t is the time (in seconds). What is the
amplitude and period of a spring in this model? y = 0.5 cos 6 t
Application #3:
(Note: Frequency of a periodic function’s graph is the reciprocal of the period. An oscillating motion
with maximum displacement a and frequency f can be modeled by:
y = a cos 2πf t)
A tuning fork vibrates with a frequency of 220 hertz (cycles per second). You strike the tuning fork
with a force that produces a maximum pressure of 3 pascals. Write a sine model that gives the
pressure P as a function of the time t (in seconds). What is the period of the sound wave?
Recall you can get the inverse of a function by switching the x and y in the function.
What is the inverse of y = sin x ? of y = cos x ?
–1
x = sin y is written y = Arcsin x or y = Sin x
–1
x = cos y is written y = Arccos x or y = Cos x
For the inverses to be functions, we must restrict the ranges. We can see this by looking at the
graphs.
Do not get
y = (sin x)
–1
confused with
y = Sin
–1
x
–1
The first is equivalent to y = csc x, while the second is an inverse function. Writing Sin x means “an
angle whose sine is x.”