Start Up Day 38
1. Solve 2cos4x - 3 = -4
over the interval
2. Solve:
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Sum and Difference Formulas
OBJECTIVE: SWBAT demonstrate and
understanding for uses of the sum and difference,
half-angle, and double-angle formulas for sine,
cosine, and tangent.
EQ: How can the sum and difference formulas
determine exact values, verify identities and
help solve trigonometric equations?
CLASS WORK: p. 425 #3, 4, 11, 15, 27, 47
&53
HOMELEARNING:http://youtu.be/oGNtanYvPpQ
p.425‐427 # 8, 12, 13, 16, 22, 28, 34, 40, 48, 54
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Sum and Difference Formulas
In this section, we will study the uses of several trigonometric
identities and formulas. (Notice that you have the condensed
version on your “shower” sheet.)
Example 1 shows how sum and difference formulas can be
used to find exact values of trigonometric functions involving sums
or differences of special angles.
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Example 1 – Evaluating a Trigonometric Function
Find the exact value of
.
Solution:
To find the exact value of
, use the fact that (
Sometimes it is helpful to convert to degrees. Think of
15°…then think of two special angles from the unit circle
that either add or subtract to build 15°…60°-45° will
work!
Consequently, the formula for sin(u – v) yields
(You may also choose to write this in degree form)
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Example 1 – Solution
cont’d
Expand by applying the formula
Replace with the exact
Unit Circle values
Multiply & re-write with
The common denominator.
Try checking this result on your calculator. You will find that
 0.259.
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Using Sum and Difference Formulas
Example 2 shows how to use a difference formula to prove
OR VERIFY the co-function identity
= sin x.
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Example 2 – Proving a Cofunction identity
Prove the co-function
identity
= sin x.
Solution:
Using the formula for cos(u – v), you have
Expand
Replace
Values
Simplify
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Using Sum and Difference Formulas
Sum and difference formulas can be used to rewrite
expressions such as
and
,
where n is an integer
as expressions involving only sin  or cos  .
The resulting formulas are called reduction formulas.
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Example 3 – Solving a Trigonometric Equation
Find all solutions of
in the interval
.
Solution:
Using sum and difference formulas, expand the equation
as
Combine like terms
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Example 3 – Solution
cont’d
æ 2ö
Replace with exact values 2sin x ç
÷ = -1
è 2 ø
2 sin x = -1
Simplify, if possible
Divide
Identify the angles on the UNIT Circle…..
So, the only solutions in the interval
are
and
.
Finally—Remember that you can check by graphing!
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