Identities with Inverse Trig Functions

[Pages:44]Identities with Inverse Trig Functions

Jeff Hicks Apr. 17, 2019

UC Berkeley

Jeff Hicks (UC Berkeley)

Identities with Inverse Trig Functions

Apr. 17, 2019 1 / 24

Summary

? Review: "Inverse" trig functions. ? Identies: Compositions of sin() and sin-1(y ). ? Example: Inverting functions with terms from trig ? Trig Identities: Right angle Identities ? Trig Identities: Even and Oddness

Jeff Hicks (UC Berkeley)

Identities with Inverse Trig Functions

Apr. 17, 2019 2 / 24

Review: Inverse Trig Functions

Review: Inverse Trig Functions

Definitions with a bit more meaning

Definition The function arcsine, which is written as sin-1(y )

? inputs a number y between -1 and 1,

?

outputs

the

angle

-

2

2

whose

vertical-coordinate

on

the

unit

circle is y .

Jeff Hicks (UC Berkeley)

Identities with Inverse Trig Functions

Apr. 17, 2019 3 / 24

Review: Inverse Trig Functions

Definitions with a bit more meaning

Definition The function arcsine, which is written as sin-1(y )

? inputs a number y between -1 and 1,

?

outputs

the

angle

-

2

2

whose

vertical-coordinate

on

the

unit

circle is y .

Definition The function arccosine, which is written as cos-1(x)

? inputs a number x between 0 and 1, ? outputs the angle 0 whose horizontal coordinate on the unit

circle is x.

Jeff Hicks (UC Berkeley)

Identities with Inverse Trig Functions

Apr. 17, 2019 3 / 24

Review: Inverse Trig Functions

Definition The function arctangent, which is written as tan-1(y )

? inputs a number m between - and , ? outputs the angle which represents a line of slope m.

Jeff Hicks (UC Berkeley)

Identities with Inverse Trig Functions

Apr. 17, 2019 4 / 24

Review: Inverse Trig Functions

What do they look like?

sin-1(y ) 1

cos-1 (x ) 2

-1 0 -1

y

1

1

tan-1(m) 1

-1 0

x 1

-5 -4 -3 -2 -1 0 -1

m

1

2

3

4

5

Jeff Hicks (UC Berkeley)

Identities with Inverse Trig Functions

Apr. 17, 2019 5 / 24

Cancelling out two functions

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