3.6 Inverse Circular Functions

Okay Gang! I get the pleasure of explaning to you the joys of Inverse Circular Funtions. Get ready for a mystical adventure!

Arccosine
The Arccosine is the same as (+/-) cos-1x. it includes all arcs whose cosine is the given numer, x, and it gives you all the other values that the calculator does not tell you. Its equation is

Arccos x = ± cos-1 + 2πn

“n” determines the amount of periods that you wish to calculate. For every value you put in place of n, you move left or right that many periods. N must be an integer, but it can be negative. A negative n value means a movement along the x axis.

The "± cos-1" portion is also known as the principal value of the relation. this is the value the calculator gives you.


Using a Graph
A sinusoidal graph can be used to determine the value(s) of the Arccos. Simply plot the sinusoid…

y= cos 2x + 1

and a line at which value you would like to calculate the arccos.



y = .9

to visually get the values for Arccosine.



Each point where the two lines intersect represent an arccos value.

Solving Algebraically
If you are given two equations, one cosine and one line, you can algebraically solve for the arccosine value. Simply put the two equations into the same equation, and solve.

Sample Problem
This sample problem relates to algebraically solving for the arccosine value.

Y1 = 9 + 7 cos (2π/13)(x-4)
Y2= 5

To slove, make both sides equal.

5 = 9 + 7 cos (2π /13)(x-4)

Subtract 9 from both sides.

-4 = 9 + 7 cos (2π/13)(x-4)

Divide both sides by seven.

( -4 / 7) = cos (2π/13)(x-4)

Use the Arccosine to get rid of the cos.

Arccos (-4/7) = (2π/13)(x-4)

“Arccos” can be replaced with “± cos-1 + 2πn”

± cos-1 ( -4/7 ) + 2π n = ( 2π / 13)( x-4)

Multiply both sides by the reciprocal of ( 2π / 13).

± ( 13 / 2π ) [cos-1 ( -4/7 ) + 2π n] = ( x-4)

Distribute the ( 13 / 2π )

± ( 13 / 2π ) cos-1 ( -4/7 ) + 13n = x-4

Add four.

± ( 13 / 2π ) cos-1 ( -4/7 ) + 13n + 4 = x

Now that you have the simplified expression, plug into your calculator and solve. Make sure you are in radian mode!

8.508, -.508

Additional Recources

http://www.pinkmonkey.com/studyguides/subjects/trig/chap6/t0606101.asp


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