Aim: Graphs of Reciprocal Functions Course: Alg. 2 & Trig.

Do Now:

Aim: How can we graph the reciprocal trig functions using the three basic trig ones?

In the diagram below of right triangle JMT, JT = 12, JM = 6 and mJMT = 90. What is the value of cot J?

J

M T

31) 3) 3

3

2 32) 2 4)

3

Aim: Graphs of Reciprocal Functions Course: Alg. 2 & Trig.

Reciprocal Identities

csc 1

sin

sec 1

cos

cot 1

tan

1sin

csc

1cos

sec

1tan

cot

Co-

Co-

Co-

Aim: Graphs of Reciprocal Functions Course: Alg. 2 & Trig.

Trig Values in Coordinate Planey

Quadrant IQuadrant II

function functionreciprocal reciprocal

x

sec is +csc is +cot is +

sec is +csc is –cot is –

cos is +sin is +tan is +

cos is +sin is –tan is –

Quadrant IVQuadrant III

cos is –sin is +tan is –

cos is –sin is –tan is +

For any given angle, a trig function and itsreciprocal have values with the same sign.

sec is –csc is –cot is +

sec is –csc is +cot is –

Aim: Graphs of Reciprocal Functions Course: Alg. 2 & Trig.

Reciprocals – Graph of Cosecant

1If 1 then 1x

x If sin 1 then csc 1x x

reciprocal of 0 - undefinedtherefore

when sin 0; csc is undefinedx x

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-2.5

-3 -2 -1 1 2 3 4 5 6

f x = sin x

when sin 1; csc =1x xwhen sin 1; csc = 1x x

these are the only points of equalityf(x) = csc x

| Real numbers and Domain =

for integral values of

Range = | 1

x x

x k k

y y

22

3

2

3

2

2

Aim: Graphs of Reciprocal Functions Course: Alg. 2 & Trig.

Reciprocals – Graph of Secant

1If 1 then 1x

x If cos 1 then sec 1x x

reciprocal of 0 undefinedtherefore

when cos 0; sec is undefinedx xwhen cos 1; sec =1x x

when cos 1; sec = 1x x these are the only points of equality

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-2.5

-3 -2 -1 1 2 3 4 5 6

g x = cos x

f(x) = sec x

2

23

2

2

3

2

| Real numbers and Domain = (2 1)

for integral values of

Range = | 1

x x

kx k

y y

Aim: Graphs of Reciprocal Functions Course: Alg. 2 & Trig.

Reciprocals – Graph of Cotangent

when tan 0; cot is undefinedx x

when tan 1; cot = 1x x the only points of equality

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-2.5

-3 -2 -1 1 2 3 4 5 6

g x = tan x

2

22

when tan is undefined; cot = 0x x

| Real numbers and Domain =

for integral values of

Range = | Real numbers

x x

x k k

y y

f(x) = cot x

3

2

-

Aim: Graphs of Reciprocal Functions Course: Alg. 2 & Trig.

Model Problems

Which expression represents the exact value of csc 60o?

3 3 2 3A. B. C. D. 3

3 2 3

Which expression gives the correct values of csc 60o?

0 1 11 0 0A. sin 60 B. sin60 C. cos60

Which is NOT an element of the domain of y = cot x?

3A. 0 B. C. D. -

2 2 2

Aim: Graphs of Reciprocal Functions Course: Alg. 2 & Trig.

Model Problems

A handler of a parade balloon holds a line of length y. The length is modeled by the function y = d sec , where d is the distance from the handler of the balloon to the point on the ground just below the balloon, and is the angle formed by the line and the ground. Graph the function with d = 6 and find the length of the line needed to form an angle of 60o.

1 66sec 6

cos cosy

6 612 feet

1cos602

y

Aim: Graphs of Reciprocal Functions Course: Alg. 2 & Trig.

Model Problem

Graph the function 2sec 3 26

y x

|a| = amplitude (vertical stretch or shrink)

2 period (when is in radians and > 0)x b

b

h = phase shift, or horizontal shift

k = vertical shift

|b| = frequency

dilation frequency phase shift vertical shift

a = 2 b = 3 k = -26

h

2

3

Aim: Graphs of Reciprocal Functions Course: Alg. 2 & Trig.

Model Problem

Graph the function 2sec 3 26

y x

3

2

1

-1

-2

-3

-4 -2 2 4 6 8

2

3

2

2

2

dilation frequency phase shift vertical shift

a = 2 b = 3 k = -26

h

6

4

2

-2

-4

-6

-8

-5 5 10 15 20

2

3period