14-3 - Militant Grammarian Download

Transcript
Fundamental
FundamentalTrigonometric
14-3Identities
14-3
Trigonometric Identities
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
Fundamental Trigonometric
14-3 Identities
Warm Up
Simplify.
1.
2.
Holt Algebra 2
cos A
1
Fundamental Trigonometric
14-3 Identities
Objective
Use fundamental trigonometric
identities to simplify and rewrite
expressions and to verify other
identities.
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
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.
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
A derivation for a Pythagorean identity is
shown below.
x2 + y2 = r2
Pythagorean Theorem
Divide both sides by r2.
cos2 θ + sin2 θ = 1
Holt Algebra 2
Substitute cos θ for
sin θ for
and
Fundamental Trigonometric
14-3 Identities
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.
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
Example 1A: Proving Trigonometric Identities
Prove each trigonometric identity.
Choose the right-hand side
to modify.
Reciprocal identities.
Simplify.
Ratio identity.
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
Example 1B: Proving Trigonometric Identities
Prove each trigonometric identity.
1 – cot θ = 1 + cot(–θ)
Choose the right-hand side
to modify.
Reciprocal identity.
Negative-angle identity.
Holt Algebra 2
= 1 + (–cotθ)
Reciprocal identity.
= 1 – cotθ
Simplify.
Fundamental Trigonometric
14-3 Identities
Helpful Hint
You may start with either side of the given
equation. It is often easier to begin with the
more complicated side and simplify it to match
the simpler side.
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
Check It Out! Example 1a
Prove each trigonometric identity.
sin θ cot θ = cos θ
cos θ
cos θ = cos θ
Holt Algebra 2
Choose the left-hand side
to modify.
Ratio identity.
Simplify.
Fundamental Trigonometric
14-3 Identities
Check It Out! Example 1b
Prove each trigonometric identity.
1 – sec(–θ) = 1 – secθ
Choose the left-hand side
to modify.
Reciprocal identity.
Negative-angle identity.
Reciprocal Identity.
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
You can use the fundamental trigonometric
identities to simplify expressions.
Helpful Hint
If you get stuck, try converting all of the
trigonometric functions to sine and cosine
functions.
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
Example 2A: Using Trigonometric Identities to
Rewrite Trigonometric Expressions
Rewrite each expression in terms of cos θ,
and simplify.
sec θ (1 – sin2θ)
Substitute.
Multiply.
cos θ
Holt Algebra 2
Simplify.
Fundamental Trigonometric
14-3 Identities
Example 2B: Using Trigonometric Identities to
Rewrite Trigonometric Expressions
Rewrite each expression in terms of sin θ, cos θ,
and simplify.
sinθ cosθ(tanθ +
cotθ)
Substitute.
Multiply.
sin2θ + cos2θ
1
Holt Algebra 2
Simplify.
Pythagorean identity.
Fundamental Trigonometric
14-3 Identities
Check It Out! Example 2a
Rewrite each expression in terms of sin θ, and
simplify.
Pythagorean identity.
Factor the difference of two squares.
Simplify.
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
Check It Out! Example 2b
Rewrite each expression in terms of sin θ, and
simplify.
cot2θ
csc2θ – 1
Pythagorean identity.
Substitute.
Simplify.
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
Example 3: Physics Application
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 sinθ = μmg cosθ
Holt Algebra 2
sinθ = μcosθ
Divide both sides by mg.
sinθ = 0.62 cosθ
Substitute 0.62 for μ.
Fundamental Trigonometric
14-3 Identities
Example 3 Continued
Divide both sides by cos θ.
tanθ = 0.62
θ = 32°
Ratio identity.
Evaluate inverse tangent.
The wooden block will start to move when
the concrete incline is raised to an angle of
about 32°.
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
Check It Out! Example 3
Use the equation mg sinθ = μmg cosθ 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 sinθ = μmg cosθ
Holt Algebra 2
sinθ = μcosθ
Divide both sides by mg.
sinθ = 0.4 cosθ
Substitute 0.4 for μ.
Fundamental Trigonometric
14-3 Identities
Check It Out! Example 3 Continued
Divide both sides by cos θ.
tanθ = 0.4
θ = 22°
Ratio identity.
Evaluate inverse tangent.
The wooden block will start to move when
the concrete incline is raised to an angle of
about 22°.
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
Lesson Quiz: Part I
Prove each trigonometric identity.
1. sinθ secθ =
2. sec2θ = 1 + sin2θ sec2θ
= 1 + tan2θ
= sec2θ
Holt Algebra 2
Fundamental Trigonometric
14-3 Identities
Lesson Quiz: Part II
Rewrite each expression in terms of cos θ,
and simplify.
3. sin2θ cot2θ secθ
cosθ
4.
2 cosθ
Holt Algebra 2