Onward to Section 53 Well start with two diagrams What is the relationship between the three angles What is the relationship between the two chords Distance formula Square both sides and expand the binomials ID: 585394
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Slide1
Sum and difference formulas
Onward to Section
5.3Slide2
We’ll start with two diagrams:
What is the relationship between the three angles?
What is the relationship between the two chords?
Distance formula:Slide3
Square both sides and expand the binomials:Slide4
(Note the sign switch in either case.)
Cosine of a Sum or DifferenceSlide5
What about the sine of a sum or difference?
Start with a
cofunction
identity:
Our new identity!!!Slide6
(Note that the sign does not switch in either case.)
Sine of a Sum or DifferenceSlide7
Directly from our previous two identities:
Tangent of a Sum or Difference
Or, use a new formula that is written entirely in terms of tangent:Slide8
Use a sum or difference identity to find an exact value of thegiven expression.
Guided Practice
1.
Any other way to simplify
this expression?
Any way to check our work
with a calculator?Slide9
Use a sum or difference identity to find an exact value of thegiven expression.
Guided Practice
2.Slide10
Use a sum or difference identity to find an exact value of thegiven expression.
Guided Practice
3.Slide11
Use a sum or difference identity to find an exact value of thegiven expression.
Guided Practice
4.Slide12
Use a sum or difference identity to find an exact value of thegiven expression.
Guided Practice
5.Slide13
Some problems require us to simply recognize certain trig.
identities…
Write the given expressions as the sine, cosine, or tangent of an
angle.
This is the sine of a sum formula!!!
This is the cosine of a
difference formula!!!Slide14
Some problems require us to simply recognize certain trig.
identities…
Write the given expressions as the sine, cosine, or tangent of an
angle.
This is the
opposite
of the
cosine of a sum formula!!!
This is the tangent of a
difference formula!!!Slide15
Verifying a Sinusoid Algebraically
Previously, we were able to re-write expressions like
as
Now, we will find an
exact
value for this function…
In general, using the sine of a sum formula:
Here, we have:
Compare coefficients:Slide16
Verifying a Sinusoid Algebraically
Solve for
a
algebraically:
Choosing
a
to be
positive gives us:
Final Answer:
orSlide17
Verifying a Sinusoid Algebraically
Express the function as a sinusoid in the form
Compare coefficients:
Solve for
a
:
Verify graphically?Slide18
Verifying a Sinusoid Algebraically
Express the function as a sinusoid in the form
Compare coefficients:
Solve for
a
:
Verify graphically?