Chapter 6 Section 6'1 Identities: Pythagorean and Sum and Difference

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Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference Warm-up

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Identities Pythagorean and Sum and Difference
Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference

• State the Pythagorean identities.
• Simplify and manipulate expressions containing
  trigonometric expressions.
• Use the sum and difference identities to find
  function values.

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Basic Identities
Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference

• An identity is an equation that is true for all
  possible replacements of the variables.

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Basic Identities
Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference
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Pythagorean Identities
Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference

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Example
Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference

• Multiply and simplify
• a)
• Solution

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Example continued
Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference

• b) Factor and simplify
• Solution

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Another Example
Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference

• Simplify the following
• trigonometric expression

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Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference

• Solution

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Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference
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Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference Warm-up
Simplify 1) 2) 3)
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Sum and Difference Identities
Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference

• There are six identities here.

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Example
Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference

• Find sin 75? exactly.

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Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference
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6.2 Identities Cofunction, Double-Angle, and
Half-Angle
Chapter 6 Section 6.2 Identities Cofunction,
Double-Angle, and Half-Angle

• Use cofunction identities to derive other
  identities.
• Use the double-angle identities to find function
  values of twice an angle when one function value
  is known for that angle.
• Use the half-angle identities to find function
  values of half an angle when one function value
  is known for that angle.
• Simplify trigonometric expressions using the
  double-angle and half-angle identities.

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Cofunction Identities
Chapter 6 Section 6.2 Identities Cofunction,
Double-Angle, and Half-Angle

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Cofunction Identities
Chapter 6 Section 6.2 Identities Cofunction,
Double-Angle, and Half-Angle

• Cofunction Identities for the Sine and Cosine

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Example
Chapter 6 Section 6.2 Identities Cofunction,
Double-Angle, and Half-Angle

• Find an identity for
• Solution

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Double-Angle Identities
Chapter 6 Section 6.2 Identities Cofunction,
Double-Angle, and Half-Angle

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Example
Chapter 6 Section 6.2 Identities Cofunction,
Double-Angle, and Half-Angle

• Find an equivalent expression for cos 3x.
• Solution

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Half-Angle Identities
Chapter 6 Section 6.2 Identities Cofunction,
Double-Angle, and Half-Angle

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Example
Chapter 6 Section 6.2 Identities Cofunction,
Double-Angle, and Half-Angle

• Find sin (? /8) exactly.
• Solution

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Another Example
Chapter 6 Section 6.2 Identities Cofunction,
Double-Angle, and Half-Angle

• Simplify .
• Solution

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Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference
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Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference
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6.4 - Inverses of the Trigonometric Functions
Chapter 6 Section 6.4 Inverses of the
Trigonometric Functions

• Find values of the inverse trigonometric
  functions.
• Simplify expressions such as sin (sin1 x) and
  sin1 (sin x).
• Simplify expressions involving compositions such
  as sin (cos1 ) without using a calculator.
• Simplify expressions such as sin arctan (a/b) by
  making a drawing and reading off appropriate
  ratios.

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Inverse Trigonometric Functions
Chapter 6 Section 6.4 Inverses of the
Trigonometric Functions
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Example
Chapter 6 Section 6.4 Inverses of the
Trigonometric Functions

• Find each of the following
• a) Find ? such that
• ? would represent a 60 or 120 angle.

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Chapter 6 Section 6.4 Inverses of the
Trigonometric Functions

• Find each of the following
• b) Find ? such that
• 
  ? would represent a 30 reference angle in
  the 2nd and 3rd quadrants. Therefore, ? 150 or
  210

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Example
Chapter 6 Section 6.4 Inverses of the
Trigonometric Functions

• Find each of the following
• c) Find ? such that
• This means that the sine and cosine of ?
  must be opposites. Therefore, ? must be 135 and
  315.

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Domains and Ranges
Chapter 6 Section 6.4 Inverses of the
Trigonometric Functions
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Composition of Trigonometric Functions
Chapter 6 Section 6.4 Inverses of the
Trigonometric Functions

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Examples
Chapter 6 Section 6.4 Inverses of the
Trigonometric Functions

• Simplify
• Since ?1/2 is in the domain of sin1,

• Simplify
• Since is not in the
• domain of cos1, does not exist.

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Special Cases
Chapter 6 Section 6.4 Inverses of the
Trigonometric Functions

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Examples
Chapter 6 Section 6.4 Inverses of the
Trigonometric Functions

• Simplify
• Since ?/2 is in the range of sin1,

• Simplify
• Since ?/3 is in the range of tan1,

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More Examples
Chapter 6 Section 6.4 Inverses of the
Trigonometric Functions

• Simplify
• Solution

• Simplify
• Solution

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Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference
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Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference
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Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference
22 / Finish Activity
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6.5 Solving Trigonometric
Equations
Chapter 6 Section 6.5 Solving Trigonometric
Equations

• Solve trigonometric equations.

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Solving Trigonometric Equations
Chapter 6 Section 6.5 Solving Trigonometric
Equations

• Trigonometric Equationan equation that contains
  a trigonometric expression with a variable.
• To solve a trigonometric equation, find all
  values of the variable that make the equation
  true.

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Example
Chapter 6 Section 6.5 Solving Trigonometric
Equations

• Solve 2 sin x ? 1 0.
• Solution First, solve for sin x on the unit
  circle.

• The values ?/6 and 5?/6 plus any multiple of 2?
  will satisfy the equation. Thus the solutions are
  
• where k is any integer.

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Graphical Solution
Chapter 6 Section 6.5 Solving Trigonometric
Equations

• We can use either the Intersect method or the
  Zero method to solve trigonometric equations. We
  graph the equations y1 2 sin x ? 1 and y2 0.

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Another Example
Chapter 6 Section 6.5 Solving Trigonometric
Equations

• Solve 2 cos2 x ? 1 0.
• Solution First, solve for cos x on the unit
  circle.

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Chapter 6 Section 6.5 Solving Trigonometric
Equations

• Solve 2 cos2 x ? 1 0.

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Graphical Solution
Chapter 6 Section 6.5 Solving Trigonometric
Equations

• Solve 2 cos2 x ? 1 0.
• One graphical solution shown.

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One More Example
Chapter 6 Section 6.5 Solving Trigonometric
Equations

• Solve 2 cos x sec x 0
• Solution

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Chapter 6 Section 6.5 Solving Trigonometric
Equations

• Solve 2 cos x sec x 0

• Since neither factor of the equation can equal
  zero, the equation has no solution.

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Graphical Solution
Chapter 6 Section 6.5 Solving Trigonometric
Equations

• 2 cos x sec x

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Last Example
Chapter 6 Section 6.5 Solving Trigonometric
Equations

• Solve 2 sin2 x 3sin x 1 0.
• Solution First solve for sin x on the unit
  circle.

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Chapter 6 Section 6.5 Solving Trigonometric
Equations

• Solve 2 sin2 x 3sin x 1 0.

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Last Example continued
Chapter 6 Section 6.5 Solving Trigonometric
Equations

• One Graphical Solution

Where k is any integer.
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Chapter 6 Section 6.1 Identities Pythagorean
and Sum and Difference
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and Sum and Difference
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