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8.3
TRIGONOMETRIC FUNCTIONS:
RELATIONSHIPS AND GRAPHS
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally
Relationships Between the Graphs
of Sine and Cosine
Example 1
Use the fact that the graphs of sine and cosine are horizontal shifts of
each other to find relationships between the sine and cosine functions.
Solution
The figure suggests that the
y = sin t
graph of y = sin t is the graph of
y = cos t
y = cos t shifted right by π/2
radians (or by 90◦). Likewise, the
graph of y = cos t is the graph of
y = sin t shifted left by π /2
radians. Thus,
sin t = cos(t − π/2)
cos t = sin(t + π/2)
The graph of y = sin t can be obtained by shifting
the graph of y = cos t to the right by π/2
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally
Relationships Involving The
Tangent Function
sin 
tan  
cos 
for
cos   0
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally
Relationships Involving Reciprocals of the
Trigonometric Functions
The reciprocals of the trigonometric functions
are given special names. Where the
denominators are not equal to zero, we
define
secant θ = sec θ = 1/cos θ.
cosecant θ = csc θ = 1/sin θ.
cotangent θ = cot θ = 1/tan θ = cos θ/sin θ.
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally
Relationships Between the Graphs
of Secant and Cosine
Example 3
Use a graph of g(θ) = cos θ to explain the shape of the graph of f(θ) = sec θ.
Solution
The figure shows the graphs of cos θ and
sec θ. In the first quadrant cos θ
decreases from 1 to 0, so sec θ increases
from 1 toward +∞. The values of cos θ
are negative in the second quadrant and
decrease from 0 to −1, so the values of
g(θ) = cos θ
sec θ increase from −∞ to −1. The graph
of y = cos θ is symmetric about the
vertical line θ = π, so the graph of f(θ) =
f(θ) = sec θ
sec θ is symmetric about the same line.
Since sec θ is undefined wherever cos θ
= 0, the graph of f(θ) = sec θ has vertical
asymptotes at θ = π/2 and θ = 3π/2.
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally
Relationships Between the Graphs
of Secant and Cosine
The graphs of y = csc θ and y = cot θ are obtained in a similar fashion
from the graphs of y = sin θ and y = tan θ, respectively.
Plots of y = csc θ and y = sin θ
Plots of y = cot θ and y = tan θ
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally
The Pythagorean Identity
We now see an extremely important relationship
between sine and cosine. The figure suggests that no
matter what the value of θ, the coordinates of the
corresponding point P satisfy the following condition:
x2 + y2 = 1. But since x = cos θ and y = sin θ, this means
cos2 θ + sin2 θ = 1
y
1
θ
x
● P = (x, y) = (cos θ, sin θ)
y
x
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally
Summarizing the
Trigonometric Relationships
• Sine and Cosine functions
sin t = cos(t − π/2) = −sin(−t)
cos t = sin(t + π/2) = cos(−t)
• Pythagorean Identity
cos2 θ + sin2 θ = 1
• Tangent and Cotangent
tan θ = cos θ/sin θ and cot θ = 1/tan θ
• Secant and Cosecant
sec θ = 1/cos θ and csc θ = 1/sin θ
Functions Modeling Change:
A Preparation for Calculus,
4th Edition, 2011, Connally