Download Holt McDougal Algebra 2

Fundamental
FundamentalTrigonometric
Identities
Trigonometric Identities
HoltMcDougal
Algebra 2Algebra 2
Holt
Fundamental Trigonometric
Identities
Objective
Use fundamental trigonometric
identities to simplify and rewrite
expressions and to verify other
identities.
Holt McDougal Algebra 2
Fundamental Trigonometric
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 McDougal Algebra 2
Substitute cos θ for
sin θ for
and
Fundamental Trigonometric
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 McDougal Algebra 2
Fundamental Trigonometric
Identities
Other versions of the Pythagorean identities may
be used also.
Look at
cos2 θ + sin2 θ = 1 and solve for cos2 θ
This should leave you with: cos2 θ = 1 - sin2 θ
Do this again this time solving for sin2 θ.
This should leave you with: sin2 θ = 1 - cos2 θ
Holt McDougal Algebra 2
Fundamental Trigonometric
Identities
Prove each trigonometric identity.
Choose the right-hand side
to modify.
Reciprocal identities.
Simplify.
Ratio identity.
Holt McDougal Algebra 2
Fundamental Trigonometric
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.
If you get stuck, try converting all of the
trigonometric functions to sine and cosine
functions.
Holt McDougal Algebra 2
Fundamental Trigonometric
Identities
Prove each trigonometric identity.
sin θ cot θ = cos θ
cos θ
cos θ = cos θ
Holt McDougal Algebra 2
Choose the left-hand side
to modify.
Ratio identity.
Simplify.
Fundamental Trigonometric
Identities
Prove each trigonometric identity.
sec θ (1 – sin2θ)=cos θ
Substitute.
Multiply.
cos θ
Holt McDougal Algebra 2
Simplify.
Fundamental Trigonometric
Identities
Verify.
sinθ cosθ(tanθ + cotθ) =1
Substitute.
Multiply.
sin2θ + cos2θ
1
Holt McDougal Algebra 2
Simplify.
Pythagorean identity.
Fundamental Trigonometric
Identities
Verify
=
Pythagorean identity.
Factor the difference of two squares.
Simplify.
Holt McDougal Algebra 2
Fundamental Trigonometric
Identities
Other strategies you may need to use are:
1. Get a common denominator.
2. Split up a fraction that has a monomial in the
denominator.
3. Distribute.
4. Simplify complex fractions.
Holt McDougal Algebra 2
Fundamental Trigonometric
Identities
Simplify
sin2θ + cos2 θ + tan2 θ
Holt McDougal Algebra 2
Fundamental Trigonometric
Identities
Verify
sin  cos 
 1 cot 
sin
Holt McDougal Algebra 2
Fundamental Trigonometric
Identities
Cwk
Pg. 359 17-25 odd
Hwk
Pg. 359 27-31 odd, 32-37
Holt McDougal Algebra 2