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.
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:
sin^{–1}(
Next, type "1," "÷," "4," ")," and "Enter" to get
Example 1
What is the measure of the angle x in 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?
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:
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