Transcript MTH 112 Elementary Functions

MTH 112
Elementary Functions
Chapter 6
Trigonometric Identities, Inverse Functions, and
Equations
Section 1
Identities: Pythagorean and Sum and Difference
Statements in Mathematics
Conditional
– May be true or false, depending on the values of the variables.
– Example: 2x + 3y = 12
Fallacy
– Never true, regardless of the values of the variables.
– Example: x = x + 1
Identity
– Always true, regardless of the values of the variables.
– Example: x2 ≥ 0
Identities from Chapter 5
Reciprocal Relationships
Tangent & Cotangent in
terms of Sine and Cosine
Cofunction Relationships
– Note: For degrees,
replace /2 with 90
Even & Odd Functions
sec x 
1
cos x
tan x 


cos x  sin   x 
2

cos(  x)  cos x
sec(  x)  sec x
csc x 
sin x
cos x
1
sin x
cot x 
cot x 


cot x  tan   x 
2

sin(  x)   sin x
csc(  x)   csc x
1
tan x
cos x
sin x


csc x  sec  x 
2

tan(  x)   tan x
cot(  x)   cot x
Pythagorean Identities
What is known about
the relationship
between x, y and ?
• x = cos 
• y = sin 

(x, y)
1
Unit Circle: x2 + y2 = 1
Pythagorean Identities
What does this
imply about the
relationship
between sin  and
cos ?
(cos , sin )

1
cos2 + sin2 = 1
Unit Circle: x2 + y2 = 1
Note: cos2 = [cos ]2
and
cos 2 = cos (2)
Pythagorean Identities
cos2x + sin2x = 1
Dividing by cos2x gives …
• 1 + tan2x = sec2x
Dividing by sin2x gives …
• cot2x + 1 = csc2x
You should also recognize any variation of these.
• example: sin2x = 1 - cos2x
Sum & Difference Formulas
7/12
= 9/12 - 2/12
= 3/4 - /6
How can we use the known values of the trig
functions of 3/4 and /6 to determine the trig
values of 7/12?
– Example:
• cos(7/12) = cos(3/4 - /6) = ???
Sum & Difference Formulas
Find cos s in terms of u and v.
(note that s = u – v)
(cos s, sin s)
s
(cos v, sin v)
B
(cos u, sin u) A
u
A
s
v
B (1, 0)
AB  2  2(cos u cos v  sin u sin v)
AB  2  2 cos s
Sum & Difference Formulas
The two expressions for AB gives …
cos(u  v)  cos u cos v  sin u sin v
Substituting –v for v gives …
cos(u  v)  cos u cos v  sin u sin v
Using the cofunctions identities gives …
sin( u  v)  sin u cos v  cos u sin v
Substituting –v for v gives …
sin( u  v)  sin u cos v  cos u sin v
Sum & Difference Formulas
Back to our original example …
cos(7/12)
= cos(9/12 - 2/12)
= cos(3/4 - /6)
= cos(3/4) cos(/6) – sin(3/4) sin(/6)
= -(√2)/2 • (√3)/2 – (√2)/2 • 1/2
= -(√6)/4 – (√2)/4
= -[(√6) + (√2)]/4
Sum & Difference Formulas
Using the sum & difference formulas for sine and cosine,
similar formulas for tangent can also be established.
tan u  tan v
tan( u  v) 
1  tan u tan v
tan u  tan v
tan( u  v) 
1  tan u tan v
Sum & Difference Formulas
Summarized
sin( u  v)  sin u cos v  cos u sin v
cos(u  v)  cos u cos v  sin u sin v
tan u  tan v
tan( u  v) 
1  tan u tan v
Simplifying Trigonometric
Expressions
No general procedure! But the following will help.
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–
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–
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Know the basic identities.
Multiply to remove parenthesis.
Factor.
Change all functions to sine and/or cosine.
Combine or split fractions: (a+b)/c = a/c + b/c
Other algebraic manipulations
Know the basic identities.
Try something and see where it takes you.
If you seem to be getting nowhere, try something else!