6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Learning Goal: I will identify equivalent trigonometric relationships. Minds On: What's the Equation?
Action: Equivalent Trigonometric Functions
Consolidation: Are we Equivalent?
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Minds On
What's the Equation?
Write 4 possible equations for this graph.
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Even and Odd Functions
An even function is a function whose graph is symmetrical in the y­axis.
An odd function is a function whose graph has rotational symmetry about the origin.
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Even and Odd Functions
Is f(θ) = sin θ an even or odd function?
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Even and Odd Functions
Is f(θ) = cos θ an even or odd function?
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Even and Odd Functions
Is f(θ) = tan θ an even or odd function?
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Cofunction Identities
π
3
2
1
π
6
√3
What expressions are equivalent?
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Cofunction Identities
θ
What is the measure of the missing angle?
Then:
sin θ =
cos θ =
tan θ =
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Equivalent Expressions from the Plane
With a related acute angle, θ, in quadrant I we can find more equivalent expressions.
Here, the angle of interest is (π ­ θ).
We can say that:
sin (π ­ θ) =
cos (π ­ θ) =
tan (π ­ θ) =
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Equivalent Expressions from the Plane
Here, the angle of interest is ______.
We can say that:
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Equivalent Expressions from the Plane
Here, the angle of interest is ______.
We can say that:
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Producing Equivalent Expressions
We can produce equivalent trigonometric expressions by:
1. Horizontal Translations
by multiples of the period
by shifting sin and cos by pi/2
2. Reflecting even functions across y­axis cos x = cos ­x
3. Reflecting odd functions across x­axis and y­axis
sin x = ­ sin ­x
­ sin x = sin ­x
tan x = ­ tan ­x
­tan x = tan ­x
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Producing Equivalent Expressions
We can produce equivalent trigonometric expressions by:
4. Using the cofunction identities
sin θ = cos (π/2 ­ θ)
cos θ = sin (π/2 ­ θ)
tan θ = cot (π/2 ­ θ)
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Action
Producing Equivalent Expressions
We can produce equivalent trigonometric expressions by:
5. Using principle and related acute angles
In Quadrant II
sin (π ­ θ) = sin θ
cos (π ­ θ) = ­cos θ
tan (π ­ θ) = ­tan θ
In Quadrant III
sin (π + θ) = ­sin θ
cos (π + θ) = ­cos θ
tan (π + θ) = tan θ
In Quadrant IV
sin (2π ­ θ) = ­sin θ
cos (2π ­ θ) = cos θ
tan (2π ­ θ) = ­tan θ
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Consolidation
Are They Equivalent?
*Use the unit circle.
WILL DO WEDNESDAY
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Consolidation
Are They Equivalent?
*Use the unit circle.
WILL DO WEDNESDAY
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Consolidation
Are They Equivalent?
*Use the unit circle.
WILL DO WEDNESDAY
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6.1 ­ Equivalent Trigonometric Functions
November 15, 2016
Consolidation
Practice Questions
Pg. 392
1 ­ 3, 5, 7
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