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Chapter 5 Trigonometric Identities. Section 5.1 Fundamental Identities. Section 5.2 Verifying Identities. Section 5.3 Cos Sum and Difference. Section 5.4 Sin & Tan Sum and Dif. Section 5.5 Double-Angle Identities. Section 5.6 Half-Angle Identities. Section 5.1 Fundamental Identities. - PowerPoint PPT Presentation

Transcript of Section 5.1 Fundamental Identities

• Section 5.1 Fundamental IdentitiesSection 5.2 Verifying IdentitiesSection 5.3 Cos Sum and DifferenceSection 5.4 Sin & Tan Sum and DifSection 5.5 Double-Angle Identities Chapter 5 Trigonometric IdentitiesSection 5.6 Half-Angle Identities

• Section 5.1 Fundamental IdentitiesReview of basic IdentitiesNegative-Angle IdentitiesFundamental Identities

• sin = cos = tan = Aopposite side = yHypotenuse = radjacent side = x

• csc = sec = cot = Aopposite side = yHypotenuse = radjacent side = xBC

• The Reciprocal Identities

sin = csc =

cos = sec =

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

• The quotient Identities

tan = =

cot = =

cos sin

• The Negative-Angle Identities

sin(-) = - sin cos(-) = cos tan(-) = - tan

• This is our firstPythagorean identity

• cos2 + sin2 1

or1 + tan2 = sec2ortan2 + 1 = sec2Pythagorean identitiescos2=cos2cos2

• cos2 + sin2 1

orcot2 + 1 = csc2or1 + cot2 = csc2Pythagorean identitiessin2sin2sin2=

• Section 5.2 Verifying IdentitiesVerify Identities by Working with One SideVerify Identities by Working with Two Sides

• Hints for Verifying IdentitiesLearn the fundamental identities and their equivalent forms.Simplify using sin and cos.Keep in mind the basic algebra applies to trig functions.You can always go down to x, y, and r

• Section 5.3 Cos Sum & DifferenceDifference Identity for CosineSum Identity for CosineCo-function IdentitiesApplying the Sum and Difference Identities

• Cosine of the Sum or Differencecos(A + B) = cos A cos B sin A sin B

cos(A - B) = cos A cos B + sin A sin B

• Co-function Identitiessin (90 - ) = cos cos (90 - ) = sin tan (90 - ) = cot csc (90 - ) = sec sec (90 - ) = csc cot (90 - ) = tan

• Section 5.4 Sine and TangentSum and Difference IdentitiesSum Identity for SineDifference Identity for SineApplying the Sum and Difference Identities for Sine

• Sine of the Sum or Differencesin(A + B) = sin A cos B + cos A sin B

sin(A - B) = sin A cos B - cos A sin B

• Tangent of the Sum or Difference

tan (A + B) =

tan (A - B) =

tan A + tan B1 tan A tan Btan A - tan B1 + tan A tan B

• Section 5.5 Double-Angle IdentitiesDouble-Angle IdentitiesVerifying Identities with Double AngelsApplying Double-Angle Identities

• Double-Angle Identity Cosinecos(2A) = cos(A+A) = cos A cos A sin A sin A = cos2 A sin2 Aorcos(2A) = cos2 A sin2 A = (1 - sin2 A) sin2 A = 1 - 2sin2 A or 2cos2 A - 1

• Double-Angle Identity Sinesin(2A) = sin(A+A) = sin A cos A + cos A sin A = 2sin A cos A

• Double-Angle Identity Tangent

tan 2A = tan (A + A) =

=

tan A + tan A1 tan A tan A 2 tan A 1 tan2A

• Section 5.6 Half-Angle IdentitiesHalf-Angel IdentitiesUsing the Half-Angle Identities

• Half-Angle Identity Sine cos 2A = 1 - 2sin2 A -cos 2A -cos 2A 0 = 1 - 2sin2 A cos 2A - 2sin2 A -2sin2 A -2sin2 A = 1 cos 2A sin2 A = (cos 2A 1) 2

• Half-Angle Identity Sine (cont.)

sin A =

sin =A2

• Half-Angle Identity Cosine cos 2A = 2cos2 A - 1 +1 +1cos 2A + 1 = 2cos2 A 2cos2 A = 1 + cos 2A cos2 A = (1 + cos 2A) 2

• Half Angle Identity Cosine (cont.)

cos A =

cos =A2

• Half-Angle Identity Tangent

tan = =

tan = A2sincosA2A2A21 cos A 1 + cos A

• Half-Angle Identity Tangent (cont)

tan = =

tan = = A2sincosA2A2A2A2A22sin cos2cos2A2sin 2 sin A1 + 2cos 1 + cos AA2( )( )A2

• Half-Angle Identity Tangent (cont)Using the other formula we get:

tan = A2