Chapter 8: Trigonometric Equations and Applications

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Chapter 8 Trigonometric Equations and
Applications
L8.4 Relationships Among the Functions
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Fundamental Trig Identities
An identity is true for all values of the
variable(s) for which each side of the equation
is defined.

• Five Sets of Fundamental Trig Identities
• Reciprocal Identities (RI)
• Quotient Identities (QI)
• Relationship with Negatives (even/odd) (EO)
• Cofunction Identities (CI)
• Pythagorean Identities (PI)

You should have the fundamental identities in
front of you when you work!
Note that the variable is the same throughout
each identity. for ex sin2 x cos2 x
1 vs. sin2 x cos2 y 1
not an identity
an identity

• The fundamental identities can be used to
• Find trig values (instead of drawing triangles)
• Simplify expressions
• Prove additional trig identities ? tomorrow
• Solve more complex trig equations ? (L8.5)

? today
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1. USING TRIG IDS TO FIND TRIG VALUES
Ex Find all trig function values if sin? ¾ and
p/2 ? lt p. Up to now, we have drawn a
triangle in Q2, solved for the missing side
using the Pyth Thm and answered the question.
Now, we can use the Pythagorean Identities
instead sin2 ? cos2 ? 1,
so cos2 ? 1 sin2?
1 (¾)2 Since
we are in Q2, cos lt 0, so cos Then
tan ? is found from the quotient ids
And the remaining trig functions are reciprocals
csc? 4/3, sec? ,
and cot
Start w/ cos
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2/3. Simplifying Trig Expressions or Proving
Identities IMPORTANT GUIDELINES
Proving or Verifying a Trig Identity 1) Work
one side only (generally the more complex side).
2) Be clear which side you have selected.

• Once youve selected a side, you are basically
  simplifying a trig expression.
• Simplifying Trig Expressions
• 1) Work the entire expression do not break
  into pieces, working each independently
  and joining them together later.
• 2) At each step, do one thing (occasionally
  two is OK). Label each step with the
  initials of the identity being used.

Generally, an expression is simplified when it
involves only one type of trig function or is a
constant. In some ways, proving identities is
easier since you know when youre done!
If your work cant be followed, its not very
compelling and you wont receive much
credit.
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2. SIMPLIFYING TRIG EXPRESSIONS
STRATEGIES 1. Use the Pythagorean identities
for expressions that match trigfcn2? 1
or trigfcn2? trigfcn2? 2. Factor 3.
Fraction Algebra - Combine fractions
- Split apart fractions 4. Multiply by
conjugates 5. Convert to sine cosine and
simplify (a strategy of last resort) An
expression is simplified when it involves only
one type of trig function or is a
constant. There are many ways to simplify
expressions. All are OK, solong as you use the
identities properly and use valid algebra.
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STRATEGY 1 Use Pythagorean Ids

2. csc2x sin2x cos2x

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STRATEGY 2 Factor

• 3. sin?cos2? sin?
• 4. cot2?csc? csc?

Note Factoring is used extensively in solving
trig equations.
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STRATEGY 3 4 Fraction Algebra / Conjugates

• COMBINE FRACTIONS (Find LCD)
• 5.
• SPLIT APART FRACTIONS
• 6.
• USE CONJUGATES
• 7.

6 7 Do not work pieces independently!There
are many ways to do these problems.
See Class Exercise p320 8
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STRATEGY 5 Convert to sine and cosine(a
strategy of last resort)

• 8. sin?sec?cot?
• 9. sec? - sin?tan?
• 10. csc?(cos3?tan? - sin?)

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OTHER EXAMPLES

• 11. cot?sin(-?)
• 12. tan?cos(-?)
• 13. 1 tan2(? p/2)

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SIMPLIFYING TRIG EXPRESSIONS
STRATEGIES 1. Use the Pythagorean identities
for expressions that match trigfcn2? 1
or trigfcn2? trigfcn2? 2. Factor 3.
Fraction Algebra - Combine fractions
- Split apart fractions 4. Multiply by
conjugates 5. Convert to sine cosine and
simplify (a strategy of last resort) An
expression is simplified when it involves only
one type of trig function or is a
constant. There are many ways to simplify
expressions. All are OK, solong as you use the
identities properly and use valid algebra.