4.2, 4.4 – The Unit Circle, Trig Functions

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4.2, 4.4 The Unit Circle, Trig Functions. The unit circle is defined by the equation x 2 + y 2 = 1. It has its center at the origin and radius 1. (0 , 1) (1 , 0) 1 (0 , 1). (1 , 0). 4.2, 4.4 The Unit Circle, Trig Functions. - PowerPoint PPT Presentation

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  • 4.2, 4.4 The Unit Circle, Trig FunctionsThe unit circle is defined by the equation x2 + y2 = 1.It has its center at the origin and radius 1.

    (0 , 1)

    (1 , 0) 1

    (0 , 1)(1 , 0)

  • 4.2, 4.4 The Unit Circle, Trig FunctionsIf the point (x , y) lies on the terminal side of , the six trig functions of can be defined as follows:

    (x , y) y xA reference triangle is made by dropping a perpendicular line segment to the x-axis.

    r2 = x2 + y2r( , +)( , )(+ , )

  • 4.2, 4.4 The Unit Circle, Trig FunctionsEvaluate the six trig functions of an angle whose terminal side contains the point (5 , 2).

    (5 , 2) 2

    5

  • 4.2, 4.4 The Unit Circle, Trig FunctionsFor a unit circle (radius 1)

    1 (1 , 0) 1(x , y)sin = y

    cos = x

    tan =

  • 4.2, 4.4 The Unit Circle, Trig Functions

    1

    (1 , 0) 1

  • 4.2, 4.4 The Unit Circle, Trig Functions

  • 4.2, 4.4 The Unit Circle, Trig FunctionsFind the six trig functions of 0

    (1 , 0)r = 1

  • 4.2, 4.4 The Unit Circle, Trig FunctionsSummary

    Deg.Rad.SinCosTan0001030451609010undef.18001027010undef.3602010

  • 4.2, 4.4 The Unit Circle, Trig Functions

    Basic Trig IdentitiesReciprocalQuotientPythagoreansin2 + cos2 = 1tan2 + 1 = sec2cot2 + 1 = csc2Cofunctionsin = cos(90 )tan = cot(90 )sec = csc(90 )Evencos() = cos sec() = sec Oddsin() = sin tan() = tan cot() = cot csc() = csc

  • 4.2, 4.4 The Unit Circle, Trig FunctionsUse trig identities to evaluate the six trig functions of an angle if cos = and is a 4th quadrant angle. sin2= 1 cos24 53

  • 4.2, 4.4 The Unit Circle, Trig FunctionsFor any angle , the reference angle for , generally written ', is always positive, always acute, and always made with the x-axis.

    '

  • 4.2, 4.4 The Unit Circle, Trig FunctionsFor any angle , the reference angle for , generally written ', is always positive, always acute, and always made with the x-axis.

    '

  • 4.2, 4.4 The Unit Circle, Trig FunctionsFor any angle , the reference angle for , generally written ', is always positive, always acute, and always made with the x-axis.

    '

  • 4.2, 4.4 The Unit Circle, Trig FunctionsFind the reference angles for and below.

    = 217 = 301

    ' = 217 180 = 37' = 360 301 = 593759

  • 4.2, 4.4 The Unit Circle, Trig FunctionsThe trig functions for any angle may differ from the trig functions of the reference angle ' only in sign.

    = 135' = 180 135 = 45sin 135= sin 45=

    =

    cos 135 = tan 135 = 1 '

  • 4.2, 4.4 The Unit Circle, Trig FunctionsA function is periodic if

    f(x + np) = f(x)

    for every x in the domain of f, every integer n, and some positive number p (called the period).

    0, 2sine & cosine period = 2 secant & cosecant period = 2tangent & cotangent period =

  • 4.2, 4.4 The Unit Circle, Trig Functionssin =

    sin =

    sin =tan =

    tan =

    tan =

  • Find the exact value of each.