6.1 B
Inverse Sine, Cosine, and Tangent
Day 2
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 4)
sin­1(sin x) = x if ­ π/2 < x < π/2
possible angles
sin(sin­1 x) = x if ­ 1 < x < 1
possible values
cos­1(cos x) = x if 0 < x < π
possible angles
cos(cos­1 x) = x if ­1 < x < 1 possible values
tan­1(tan x) = x if ­ π/2 < x < π/2
possible angles
tan(tan­1 x) = x if ­ ∞ < x < ∞
possible values
Title: Feb 4­3:21 PM (2 of 4)
Sine
First there’s the venerable sine wave. In MojoWorld we multiply sine waves along the various dimensions. As the sine function is periodic, anything built using the sine basis will be periodic. Periodic phenomena are quite common in Nature, but they tend to look unnatural in synthetic images. Nature can get away with things that we can’t. sin­1(sin x) = x if ­ π/2 < x < π/2
Are the following true?
Why or why not?
tan­1(tan π/6 ) = π/6 Yes :) sin(sin­1 x) = x if ­ 1 < x < 1
cos­1(cos 1/6) = 1/6 Yes :)
cos­1(cos x) = x if 0 < x < π
sin(sin­1 2) = 2 No, not in range
tan(tan­1 2054 ) = 2054 Yes :) cos(cos­1 x) = x if ­1 < x < 1 cos(cos­1 ­.995) = ­.995 Yes :)
sin­1(sin π/2 ) = π/2 Yes :)
tan­1(tan x) = x if ­ π/2 < x < π/2
tan(tan­1 x) = x if ­ ∞ < x < ∞
tan­1(tan π/6 ) = π/6 Yes :) cos­1(cos π/6) = π/6 Yes :)
sin(sin­1 17π/6) = 17π/6 No, not in range
Title: Feb 4­3:21 PM (3 of 4)
H.W. ­ pg 458
37 ­ 56 and 64 Complex tangent function f ( z ) = tan z. Corners at ± π ± π i
Title: Feb 4­3:21 PM (4 of 4)