Math 150 – Fall 2015
Section 8C
Section 8C – Graphs of the Trig Functions
Basic Graphs
y = sin x
y = cos x
y = tan x
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Math 150 – Fall 2015
Section 8C
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y = cot x
y = csc x
y = sec x
Note.
• Mathematicians always use radians when evaluate and graph the trig functions. This will make the derivatives of the trig functions easier in calculus.
• Since the coordinates of a point on the circle repeat with each revolution, the
graphs of the basic trig functions repeat every 2π. Therefore, the trig functions
are periodic with period 2π (or π in the case of y = tan x and y = cot x.)
• The x and y are the input and the output respectively of the function y = sin x.
When we evaluate y = sin x, the x is the angle in radians, and y is the value of
the trig function for that angle.
Math 150 – Fall 2015
Section 8C
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Caution: The x and y in y = sin x have absolutely nothing to do with the x and y
coordinates in the circle definition of the trig functions.
Definition. For transformations of the basic trig functions y = sin x and y = cos x, we
define amplitude a to be
a=
|maximum value − minimum value|
.
2
The amplitude is the maximum distance the function goes above and below the average
value, or middle value, of the function.
The period of the function is the time it takes for the function to repeat.
Theorem. The curves of y = a sin k(x − b) + c and y = a cos K(x − b) + c for k > 0
have
• Amplitude |a|
• phase shift b (right or left)
• vertical shift |b| up or down
• The period P is P =
2π
k .
Example 1. For the following functions find the amplitude, period, reflection, phase
shift, and vertical shift.
(a) y = 4 cos(3x − 6) − 4
(b) y = −5 sin(5x + 15) + 7
Math 150 – Fall 2015
Section 8C
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Example 2. Graph the function y = 4 cos(2x).
Example 3. For each of the graphs below, write a function of the form f (x) =
a sin k(x − b) + c and of the form g(x) = a cos k(x − b) + c, where a, k, and b are
positive and as small as possible.
(a)
(b)
(c)