Download File - College Algebra Fundamentals

1.6 Graphs of Other
Trigonometric Functions
Part 5
 Sums
and differences of
sinusoids with the same
period are again sinusoids.
 The
result will also have that
period.
Sums and Differences of Sinusoids
If y1  a1 sin bx  c1  and y2  a2 cosbx  c2 
then:
y1  y2  a1 sin bx  c1   a2 cosbx  c2 
 This
final function is also a sinusoid with
a period of 2π/│b│
 Works the same for the sum of 2 sine or 2
cosine functions.
 But for the sum to also be a sinusoid, the 2
individual sinusoids MUST have the SAME
period.
Examples (Identifying Sinusoids)

Which of the following are SINUSOIDS?
1. f x  5 cos x  3 sin x

Yes, Since both have a period of 2π

 
 
2. h x  2 cos 3x  3 cos 2 x
No, 2 cos 3x has a period of 2π/3 and 3 cos 2x
has a period of π.

3. g x  cos 5x  sin 3x
No, since cos 5x has a period of 2π/5 and sin 3x
has a period of 2π/3.
Examples (Identifying Sinusoids)
Continued
4.
 5x 
 5x 
 5x 
r  x   a cos   b cos   c sin  
 7 
 7 
 7 
Yes, since all three have a period of 14π/5
Now…
A Summary of the 6
basic trig functions
The Sine Function






Fact: The function and the
sinus cavities derive their
names from a common root:
the Latin word for “bay.”
Domain: ( , )
Range: [ 1,1]
Continuous? Yes
Symmetry? Yes, Odd
Period: 2π
f ( x)  sin( x)
The Cosine Function






Fact: The local extrema of
this function occur exactly
at the zeros of the sine
function, and vice versa.
Domain: (, )
Range: [1,1]
Continuous? Yes
Symmetry? Yes, Even
Period: 2π
f ( x)  cos( x)
The Tangent Function






Domain: All reals
except odd multiples
of π/2
Range: (-∞, ∞)
Continuous? Yes, on
its domain
Symmetry? Yes, Odd
Asymptotes:
Vertical: x = k(π/2) for
odd integers k
Period: π
f ( x )  tan x
The Cotangent Function





Reciprocal of the
tangent function (there
is no key for this on the
calculator).
Cot x = (cos x) / (sin x)
Its zeros are where
cos(x) = 0. (odd
multiples of π/2)
Vertical asymptotes
where sin(x) = 0.
(multiples of π)
Period = π
The Secant Function


f(x) = 1 / (cos x) = sec x
Vertical Asymptotes: When cos x = 0 (sec x is
undef.) The period of the secant function is the same
as the cosine function (2π).
y = cos(x)
y = sec(x)
Vertical asymptotes at
x

2
 n .
Range: (–∞, –1] U [1, ∞)
The Cosecant Function


f(x) = 1 / sin x = csc x
Vertical Asymptotes: When sin x = 0 (csc x is undef.)
The period of the cosecant function is the same as the
sine function (2π).
y = sin(x)
y = csc(x)
Vertical asymptotes at
x  n .
Range: (–∞, –1] U [1, ∞)
Classwork

1.6 Practice - Damped Oscillation;
Sums and Differences of Sinusoids
Book: Pg 182; 77 - 82
Homework:
Pg 178 – 180
Exercises: 1 – 8; 13, 18, and 19