# 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.