EXAMPLE 20.5: Solving a Complicated Trigonometry Problem

  EXAMPLE 20.5: Solving a Complicated Trigonometry Problem (Continued)

Notice that m ∠ ECD = m ∠ BAD because both ∠ BCD and ∠ BAD are complimentary to ∠ ADB.

Inverse Trigonometric Functions

Now that we know how to use the trigonometric functions to find the length of a side of a right triangle, we can discuss how to find an unknown angle.

The six trigonometric functions involve the ratios of the lengths of the sides of right triangles. We have seen that for a given angle, the value of a trigonometric function is the same regardless of the size of the triangle. Consequently, if we know the ratio of any two sides of a right triangle, we can operate backwards and determine the measure of an angle by using the correct inverse trigonometric function

.

For instance, we can easily show that the sin 30 ° = 0.5 using a scientific calculator. Suppose we are given a right triangle, whose opposite side is 5 inches and hypotenuse is 10 inches, and asked to find the measure of one of the acute angles, A . This amounts to asking for the angle whose sine is 5/10 = 0.5, which we know to be 30 ° . In general, we solve this type of problem using the inverse trigonometric functions as follows:

sin A = 0.5

     A = sin^-1(0.5)

     A = 30 °

sin ^–1 is the symbol for the inverse sine function. It does not mean sine raised to the negative 1 power.

Each of the six trigonometric functions has an inverse function. Sometimes this function is called the arcfunction as in arcsine, arccosine, and arctangent. Some scientific calculators use sin^–1 to denote the inverse sine while others use the INV key, or some other designation.

EXAMPLE 20.6: Finding an Unknown Angle Using Trigonometry

  In general, sin   = x   sin ^{- 1} x = .

  Notice that   + = 90 .