Fundamental

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Fundamental Trigonometric Identities
11-3
Warm Up
Lesson Presentation
Lesson Quiz
Holt Algebra 2
Holt McDougal Algebra 2
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Warm Up Simplify. 1. 2.
cos A
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Objective
Use fundamental trigonometric identities to
simplify and rewrite expressions and to verify
other identities.
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You can use trigonometric identities to simplify
trigonometric expressions. Recall that an
identity is a mathematical statement that is true
for all values of the variables for which the
statement is defined.
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A derivation for a Pythagorean identity is shown
below.
x2 y2 r2
Pythagorean Theorem
Divide both sides by r2.
cos2 ? sin2 ? 1
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To prove that an equation is an identity, alter
one side of the equation until it is the same as
the other side. Justify your steps by using the
fundamental identities.
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Example 1A Proving Trigonometric Identities
Prove each trigonometric identity.
Choose the right-hand side to modify.
Reciprocal identities.
Simplify.
Ratio identity.
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Example 1B Proving Trigonometric Identities
Prove each trigonometric identity.
Choose the right-hand side to modify.
1 cot ? 1 cot(?)
Reciprocal identity.
Negative-angle identity.
1 (cot?)
Reciprocal identity.
1 cot?
Simplify.
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Check It Out! Example 1a
Prove each trigonometric identity.
sin ? cot ? cos ?
Choose the left-hand side to modify.
cos ?
Ratio identity.
cos ? cos ?
Simplify.
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Check It Out! Example 1b
Prove each trigonometric identity.
Choose the left-hand side to modify.
1 sec(?) 1 sec?
Reciprocal identity.
Negative-angle identity.
Reciprocal Identity.
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You can use the fundamental trigonometric
identities to simplify expressions.
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Example 2A Using Trigonometric Identities to
Rewrite Trigonometric Expressions
Rewrite each expression in terms of cos ?, and
simplify.
sec ? (1 sin2?)
Substitute.
Multiply.
Simplify.
cos ?
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Example 2B Using Trigonometric Identities to
Rewrite Trigonometric Expressions
Rewrite each expression in terms of sin ?, cos ?,
and simplify.
sin? cos?(tan? cot?)
Substitute.
Multiply.
sin2? cos2?
Simplify.
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Pythagorean identity.
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Check It Out! Example 2a
Rewrite each expression in terms of sin ?, and
simplify.
Pythagorean identity.
Factor the difference of two squares.
Simplify.
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Check It Out! Example 2b
Rewrite each expression in terms of sin ?, and
simplify.
cot2?
csc2? 1
Pythagorean identity.
Substitute.
Simplify.
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Example 3 Physics Application
Use the equation mg sin? µmg cos?. At what
angle will a wooden block on a concrete incline
start to move if the coefficient of friction is
0.62?
Set the expression for the weight component equal
to the expression for the force of friction.
mg sin? µmg cos?
Divide both sides by mg.
sin? µcos?
Substitute 0.62 for µ.
sin? 0.62 cos?
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Example 3 Continued
Divide both sides by cos ?.
Ratio identity.
tan? 0.62
Evaluate inverse tangent.
? 32
The wooden block will start to move when the
concrete incline is raised to an angle of about
32.
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Check It Out! Example 3
Use the equation mg sin? µmg cos? to determine
the angle at which a waxed wood block on a wood
incline with µ 0.4 begins to slide.
Set the expression for the weight component equal
to the expression for the force of friction.
mg sin? µmg cos?
Divide both sides by mg.
sin? µcos?
Substitute 0.4 for µ.
sin? 0.4 cos?
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Check It Out! Example 3 Continued
Divide both sides by cos ?.
Ratio identity.
tan? 0.4
Evaluate inverse tangent.
? 22
The wooden block will start to move when the
concrete incline is raised to an angle of about
22.
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Lesson Quiz Part I
Prove each trigonometric identity.
1. sin? sec?
2. sec2? 1 sin2? sec2?
1 tan2?
sec2?
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Lesson Quiz Part II
Rewrite each expression in terms of cos ?, and
simplify.
3. sin2? cot2? sec?
cos?
4.
2 cos?