Inverse Functions Study Guide (page 2)

Updated on Oct 2, 2011

Example 2

Evaluate sin(sin–1(0.7)).


sin–1(0.7) = θ, where sin(θ) = 0.7


The angle θ is sketched in Figure 14.4.

Figure 14.4

The opposite side is O = 7 and the hypotenuse is H = 10 so that . The adjacent side is A = √102 – 72 = √5l by the Pythagorean theorem. Thus,



This problem could have been solved much more directly by pronouncing sin(sin–1(0.7)). This reads, "What is the sine of the angle whose sine is O.7?" Clearly, the sine of that angle is 0.7. It is in this sense that an inverse undoes a function. We might as well just cross out the sin and sin–1 from the original sin(sin–1(0.7)), leaving behind just the 0.7.

Inverse Trigonometric Functions on a Calculator

A calculator can be used to estimate inverse trigonometric functions. For example, suppose you want to know what angle θ has . Thus, we want to evaluate . Usually SIN–1 is written in small print above the "SIN" button on a calculator. You need to first press "2nd" or "INV" and then press the "SIN" button. On an inexpensive scientific calculator, press" 1,""÷," "4," and then "=" to get 0.25 on the screen. Next, press "2nd" (or "INV") and then "SIN." The result will be around 14.478° if your calculator is in degrees mode and 0.253 if your calculator is in radians mode. On a fancier graphing calculator, press "INV" first, and then "SIN" so that the screen shows:


Next, type "1," "÷," "4," ")," and "Enter" to get

Example 1

What is the measure of the angle x in Figure 14.5?

Figure 14.5

We know that . This means that . With a calculator, this comes out to approximately x ≈ 52°, or 0.908 radians.

Example 2

What is the measure of θ in Figure 14.6?

Figure 14.6

Here, we know that the opposite side is 0 = 7 feet and the adjacent side is A = 4 feet. Thus, . Equivalently, . With a calculator, this comes out to θ ≈ 60.26°, or 1.05 radians.

Practice problems for this study guide can be found at:

Inverse Functions Practice Questions

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