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Inverse Trigonometric Functions and Trigonometric Equations

This page is meant to serve as a quick overview of inverse trigonometric functions and trigonometric equations.

Inverse Trigonometric Functions

Function Domain Range Quadrants for Range Values
y = sin^(-1)x [-1,1] [-π/2,π/2] I and IV
y = cos^(-1)x [-1,1] [0,π] I and II
y = tan^(-1)x (-∞,∞) (-π/2,π/2) I and IV
y = cot^(-1)x (-∞,∞) (0,π) I and II
y = sec^(-1)x (-∞,-1] U [1,∞) [0,π/2) U (π/2,π) I and II
y = csc^(-1)x (-∞,-1] U [1,∞) [-π/2,0) U (0,π/2) I and IV

Graphs

y = sin^(-1)x or arcsin x y = cos^(-1)x or arccos x y = tan^(-1)x or arctan x
y = csc^(-1)x or arccsc x y = sec^(-1)x or arcsec x y = cot^(-1)x or arccot x

Solving Trigonometric Equations Algebraically

Step 1 Determine whether the equation is linear or quadratic.
Step 2 Determine if there is more than one trigonometric function present. If there is only one, use Step 3A. If there are more than one, use step 3B.
Step 3A Solve the equation for that function.
Step 3B Rearrange the equation so that one side equals zero. Then, factor and then set each factor equal to zero. If this cannot be done, use identities to change the form of the equations, but make sure to check for extraneous solutions.
Note: If the equation you determine to be the end result is in quadratic form, but not factorable, use the quadratic forumla. If this is done, be sure to check for extraneous solutions.

Solving Trigonometric Equations Graphically

This procedure if for any equation in the form of f(x) = 0.

Step 1 Graph y = f(x)
Step 2 Find the x-intercepts.

This procedure if for any equation in the form of f(x) = g(x).

Step 1 Graph y = f(x) and y = g(x)
Step 2 Find the x-coordinates where the two equations intersect.

This procedure if for any equation in the form of f(x) = g(x).

Step 1 Graph y = f(x) and y = g(x)
Step 2 Find the x-coordinates where the two equations intersect.

This procedure if for any equation in the form of f(x) = g(x).

Step 1 Graph y = f(x) and y = g(x)
Step 2 Find the x-coordinates where the two equations intersect.