6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Chapter 6: Trigonometric Identities
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Chapter 6
6.1 Reciprocal, Quotient, and Pythagorean Identities
Pages 290­ 298
6.3 Proving Identities Pages 309 ­ 315
Complete the following table:
Measure of <A
0o
30o
45o
60o
90o
sin A
cos A
sin2 A + cos2 A
What conclusion do you make?
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Trigonometric Identity­ a trigonometric equation that is true for all permissible values of the variable expressions on both sides of the equation.
Pythagorean Identities:
Reciprocal Identities: Quotient Identities: 3
6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
NOTE: All identities can be rearranged to suit the problem at hand.
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Example 1
Page 291
Verify a Potential Identity Numerically and Graphically
a) Determine the non­permissible values, in degrees, for the equation .
b) Numerically verify that θ = 60° and θ = are solutions of the
equation.
c) Use technology to graphically decide whether the equation could be an identity over the domain –360° < θ ≤ 360°.
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Example 1: Your Turn
Page 293
a) Determine the non­permissible values, in degrees, for the equation
b) Verify that x = 45° and x = are solutions to the equation.
Answer
c) Use technology to graphically decide whether the equation could be
an identity over the domain –360° < x ≤ 360°.
a) x ≠ 180
b) For x = For x = :
c) The gra
could be
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Example 2
Page 293
Use Identities to Simplify Expressions
a) Determine the non­permissible values, in radians, of the variable in
b) Simplify the expression.
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Example 2: Your Turn
Page 294
the expression b) Simplify the expression.
Answer
a) Determine the non­permissible values, in radians, of the variable in
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Example 3
Page 295
Use the Pythagorean Identity
a) Verify that the equation cot2 x + 1 = csc2 x is true when x =
b) Use quotient identities to prove cot2 x + 1 = csc2 x.
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
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a) Verify the equation 1 + tan2 x = sec x numerically for x = b) Use quotient identities to prove 1 + tan2 x = sec2 x.
Answer
Example 3: Your Turn
Page 295
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
More Examples: 1. Simplify the following:
a) b)
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
2. Using :
a) Verify it is true for b) Prove the identity.
c) State and non­permissible value.
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Example 1 Verify Versus Prove That an Equation Is an Identity
page 310 a) Verify that 1 – sin2 x = sin x cos x cot x for some values of x.
Determine the non­permissible values for x. Work in degrees.
b) Prove that 1 – sin2 x = sin x cos x cot x for all permissible values of x.
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Example 1: Your Turn
Page 311
b) Verify that the equation may be an identity, either graphically using technology or by choosing one value for x.
c) Prove that the identity is true for all permissible values of x.
Answer
a) Determine the non­permissible values for the equation
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Example 3
Page 312
Prove that is an identity for all permissible
values of x.
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6.1 Reciprocal, Quotient, and Pythagorean Identities.notebook
Some helpful hints when simplifying or proving identities:
­> when working with identities, keep left side and right side separate NEVER cross the equals sign and work with more complicated side of the equation
­> if there are squared terms, check to see if a Pythagorean identity can be applied.
­> express in terms of sine and cosine
­> if above does not work, multiply by an expression equivalent to 1 (same idea as rationalizing)
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