Section 5.4 Properties of Trigonometric Functions
Basic Trigonometric identities:
Note: restrict domain so that both sides of the equation are defined.
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Reciprocal Identities
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1
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csc(x) =
sec(x) =
cot(x) =
sin(x)
cos(x)
tan(x)
Quotient Identities
sin(x)
cos(x)
tan(x) =
cot(x) =
cos(x)
sin(x)
Identities for Negatives
sin("x) = "sin(x) cos("x) = cos(x) tan("x) = "tan(x)
Pythagorean Identities
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sin 2 (x) + cos2 (x) = 1
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Question: What identity do we get if we divide both sides of the Pythagorean identity by
cos2 (x) ?
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Question: What identity do we get if we divide both sides of the Pythagorean identity by
sin 2 (x) ?
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1. Graph y = sin(x)
Period:
Domain:
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Range:
Odd – Even
2. Graph y = cos(x)
Period:
Domain:
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Range:
Odd – Even
3. Graph y = tan(x)
Period:
Domain:
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Range:
Odd – Even
4. Graph y = cot(x)
Period:
Domain:
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Range:
Odd – Even
5. Graph y = sec(x)
Period:
Domain:
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Range:
Odd – Even
6. Graph y = csc(x)
Period:
Domain:
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Range:
Odd – Even
7. Graph y = Acos(x) " 2# $ x $ 2# " 3 $ y $ 3 for A = 1, 2, -3, all in the same
viewing window.
Do the x-intercepts change?
How far does each graph deviate from the x-axis? (experiment with additional values of
!A)
8. Graph y = Asin(x) " 2# $ x $ 2# " 3 $ y $ 3 for A = 1, 2, -3, all in the same
viewing window.
Do the x-intercepts change?
How far does each graph deviate from the x-axis? (experiment with additional values of
!A)
9. Graph y = cos(Bx) " # $ x $ # " 2 $ y $ 2 for B = 1, 2, 3, all in the same viewing
window.
How many periods of each graph appear in the viewing window? (Experiment with
additional values of B)
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10. Graph y = sin(Bx) " # $ x $ # " 2 $ y $ 2 for B = 1, 2, 3, all in the same viewing
window.
How many periods of each graph appear in the viewing window? (Experiment with
additional values of B)
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