Complex Analysis/Elementary Functions/Inverse Trig Functions

Solve Equations Using Inverses
Oftentimes, the value of a trigonometric function for an angle is known and the value to be found is the measure of the angle. In order to find the inverse of trigonometric functions, the idea of inverse functions is applied.

The relation in which all the values of x and y are reversed in the inverse of a function. y = sinx has an inverse of x = siny.

When graphed, the inverse x = siny is found to not be a function since it doesn't pass the vertical line test. Similarly, trigonometric inverses aren't functions either.

In order to make trigonometric inverses functions, the domain of the original trigonometric function has to be restricted. These are known as principal values. Principal values are those values that are in the restricted domains. In order to differentiate trigonometric functions with restricted domains, capital letters are used.

Principal Values of Sine, Cosine, and Tangent

y = Sinx if and only if y = sinx and -pi/2 < x < pi/2.

y = Cosx if amd only if y = cosx and 0 < x < pi.

y = Tanx if and only if y = tanx and -pi/2 < x < pi/2.

The Arcsine function is the inverse of the Sine function. It is symbolized by Sin-1 or Arcsin. These are its characteristics:

1. The set of real numbers from -1 to 1 is its domain.

2. The set of angle measures from -pi/2 < x < pi/2 is its range.

3. Sin-1y = x if and only if Sinx = y.

4. (Sin-1 x Sin)(x) = (Sin x Sin-1)(x) = x

Arccosine and Arctangent functions are similar to the above definition of the Arcsine function.

Inverse Sine, Cosine, and Tangent

1. The inverse Sine function is y = Sin-1x or y = Arcsinx given y = Sinx.

2. The inverse Cosine function is y = Cos-1x or y = Arccosx given y = Cosx.

3. The inverse Tangent function is y = Tan-1x or y = Arctanx given y = Tanx.

The expressions in rows below are all equivalent. These can be used to rewrite and/or solve trigonometric equations.

y = Sinx    x = Sin-1y     x = Arcsiny

y = Cosx    x = Cos-1y     x = Arccosy

y = Tanx    x = Tan-1y     x = Arctany

Example 1
Solve an Equation

Solve Sinx = 1/2 by finding the value of x to the nearest degree.

If Sinx = 1/2, then x is the least value whose sine is 1/2. So, x = Arcsin(1/2). Use a calculator to find x.

For a TI-84 Plus Silver Edition:

1. Press 2nd

2. Sin-1

3. 2nd

4. 1/2

5. )

6. Enter

The answer is 30. So, x = 30 degrees.

The inverse of trigonometric functions is also used in application problems.

Example 2
Apply an Inverse to Solve a Problem

''The ship Vegas sailed West 25 miles before turning south. When Vegas ran into trouble and radioed for help, the rescue boat found that the fastest way to them covered a distance of 50 miles. The cosine of the angle that the rescue boat should sail at is 0.5. Find the angle, to the nearest hundredth of a degree, at which the rescue boat should travel to give Vegas help.''

Cosx = 25/50

Cos-1(25/50) = Cos-1(0.5) = 60

60 degrees south of west

Trigonometric Values
The values of trigonometric expressions are also found using a calculator.

Example 3
Find a Trigonometric Value

''Find each value. Write angle measures in radians. Round to the nearest hundredth.''

ArcTan(1)

For TI-84 Plus Silver Edition:

1. 2nd

2. TAN-1

3. 2nd

4. 1

5. ENTER

0.7853981634

So, ArcTan(1) = 0.7853981634