6.1A
Inverse Sine, Cosine, and Tangent
Day 1
Objective: Find the exact value of Inverse Sine, Cosine & Tangent Functions
[sin(3 * phi)4 + cos(3 * phi)4 + sin(3 * theta)4 + cos(3 * theta)4] Title: Feb 4­3:21 PM (1 of 7)
y = sin­1 x means x = sin y
y is an angle x is a value
sin­1(1/2) = 60o sin(60o) = 1/2 sin­1(60o) = NO
Said "Y equals the inverse sine of X"
or "Y is the angle whose sine is X"
You can think θ = sin­1 x
Title: Feb 4­3:21 PM (2 of 7)
The sine function is not a one to one function. The inverse requires a one to one funtion. So we only take a part of the funtion.
We take the interval ­ π/2 to π/2 or ­90o to 90 o to make it one to one.
Notice that the Range is ­1 to 1.
* send the graph to the back
Title: Feb 4­3:21 PM (3 of 7)
This same idea applies to the cosine and the tangent funtions.
y = sin­1x for ­1 < x < 1 and ­π/2 < y < π/2
y = cos­1x for ­1 < x < 1 and 0 < y < π y = tan­1x for ­∞ < x < ∞ and ­π/2 < y < π/2
1
π
2π
­1
Send graph to back.
Title: Feb 4­3:21 PM (4 of 7)
Send graph to back.
Angle values
Reset
Sine
Cosine
Tangent
for ­∞ < x < ∞ for ­1 < x <1 for ­1 < x < 1 and ­ π/2 < y < π/2
and ­ π/2 < y < π/2
and 0 < y < π Title: Feb 4­3:21 PM (5 of 7)
Find the value of:
tan ­1(1) = π/4
sin ­1(­1/2) = ­ π/6
cos­1(­√3/2) = 5π/6
tan­1( ­√3/3 ) = ­ π/6
sin­1(­1) = ­ π/2
cos ­1(3π /4) = not a possible value!
tan ­1(0) = 0 or 2π
cos ­1(­1/2) = 2π/3
Title: Feb 4­3:21 PM (6 of 7)
The fifth iterate f(5)( z ) of f ( z ) = ( 1 + i ) sin z. Corners at ± 3 ± 3 i
H.W. ­ pg 457
7 ­ 36 Title: Feb 4­3:21 PM (7 of 7)