Until now we have only been dealing with the trigonometric values that we known. But what about the trigonometric values we dont know and we dont have calculators to find out? This is where the inverse of the trigonometry comes into place. The standard form for the inverse of trigonometry is y = sin(or any other trig signs)^-1 x. But we usually write this way: y = arcsin(or any other trig signs) x. These two equations are the same, and they all mean that a trigonometry value has value of y. For example: arcsin ½ = P /6 + 2P K and 5P /6 + 2P K. You have to add 2P K because arcsin goes on forever. There is also a finite version of arcsin called Arcsin with capitalized A. Arcsin is finite and is also called the principle value. Here are the limits for Arc:
y = Arctan x y = Arcsin x
y = Arccsc x y = [ -P /2, P /2 ]
y = Arccos x y = Arccot x
y = Arcsec x y = [ 0, P ]
So Arcsin ½ = P /6. We can also convert arcsin ½ to principle value to +- Arcsin ½ + 2P K.
We can use the inverse of trigonometry to solve all kinds of equations. Following is an example:
sin^3 x cos x sin x cos^3x = -1/4
sin x cos x (sin^2x cos^2x) = -1/4
2 sinx cosx cos2x = ½
(sin 2x)(cos 2x) = ½
2 sin 2x cos 2x = 1
sin 4x =1
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