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CHAPTER 5. ANALYTIC TRIGONOMETRY. 5.1 Verifying Trigonometric Identities. Objectives Use the fundamental trigonometric identities to verify identities. Fundamental Identities. Reciprocal identities csc x = 1/sin x sec x = 1/cos x cot x = 1/tan x Quotient identities

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CHAPTER 5

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Chapter 5 l.jpg

CHAPTER 5

ANALYTIC TRIGONOMETRY


5 1 verifying trigonometric identities l.jpg

5.1 Verifying Trigonometric Identities

  • Objectives

    • Use the fundamental trigonometric identities to verify identities.


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Fundamental Identities

  • Reciprocal identities

    csc x = 1/sin x sec x = 1/cos x cot x = 1/tan x

  • Quotient identities

    tan x = (sin x)/(cos x) cot x = (cos x)/(sin x)

  • Pythagorean identities


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Fundamental Identities (continued)

  • Even-Odd Identities

    • Values and relationships come from examining the unit circle

      sin(-x)= - sin x cos(-x) = cos x

      tan(-x)= - tan x cot(-x) = - cot x

      sec(-x)= sec x csc(-x) = - csc x


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Given those fundamental identities, you PROVE other identities

  • Strategies: 1) switch into sin x & cos x, 2) use factoring, 3) switch functions of negative values to functions of positive values, 4) work with just one side of the equation to change it to look like the other side, and 5) work with both sides to change them to both equal the same thing.

  • Different identities require different strategies! Be prepared to use a variety of techniques.


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Verify:

  • Manipulate right to look like left. Expand the binomial and express in terms of sin & cos


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5.2 Sum & DifferenceFormulas

  • Objectives

    • Use the formula for the cosine of the difference of 2 angles

    • Use sum & difference formulas for cosines & sines

    • Use sum & difference formulas for tangents


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cos(A-B) = cosAcosB + sinAsinBcos(A+B) = cosAcosB - sinAsinB

  • Use difference formula to find cos(165 degrees)


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sin(A+B) = sinAcosB + cosAsinBsin(A-B) = sinAcosB - cosAsinB


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5.3 Double-Angle, Power-Reducing, & Half-Angle Formulas

  • Objectives

    • Use the double-angle formulas

    • Use the power-reducing formulas

    • Use the half-angle formulas


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Double Angle Formulas(developed from sum formulas)


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You use these identities to find exact values of trig functions of “non-special” angles and to verify other identities.


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Double-angle formula for cosine can be expressed in other ways


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We can now develop the Power-Reducing Formulas.


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These formulas will prove very useful in Calculus.

  • What about for now?

  • We now have MORE formulas to use, in addition to the fundamental identities, when we are verifying additional identities.


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Half-angle identities are an extension of the double-angle ones.


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Half-angle identities for tangent


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5.4 Product-to-Sum & Sum-to-Product Formulas

  • Objectives

    • Use the product-to-sum formulas

    • Use the sum-to-product formulas


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Product to Sum Formulas


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Sum-to-Product Formulas


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5.5 Trigonometric Equations

  • Objectives

    • Find all solutions of a trig equation

    • Solve equations with multiple angles

    • Solve trig equations quadratic in form

    • Use factoring to separate different functions in trig equations

    • Use identities to solve trig equations

    • Use a calculator to solve trig equations


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What is SOLVING a trig equation?

  • It means finding the values of x that will make the equation true. (Just as we did with algebraic equations!)

  • Until now, we have worked with identities, equations that are true for ALL values of x. Now we’ll be solving equations that are true only for specific values of x.


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Is this different that solving algebraic equations?

  • Not really, but sometimes we utilize trig identities to facilitate solving the equation.

  • Steps are similar: Get function in terms of one trig function, isolate that function, then determine what values of x would have that specific value of the trig function.

  • You may also have to factor, simplify, etc, just as if it were an algebraic equation.


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Solve:


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