Section 6.3 Inverse Trig Functions - Opentextbookstore

Section 6.3 Inverse Trig Functions 379Section 6.3 Inverse Trig FunctionsIn previous sections we have evaluated the trigonometric functions at various angles, butat times we need to know what angle would yield a specific sine, cosine, or tangent value.For this, we need inverse functions. Recall that for a one-to-one function, if f (a ) b ,then an inverse function would satisfy f 1 (b) a .You probably are already recognizing an issue – that the sine, cosine, and tangentfunctions are not one-to-one functions. To define an inverse of these functions, we willneed to restrict the domain of these functions to yield a new function that is one-to-one.We choose a domain for each function that includes the angle zero.ª S SºSine, limited to « , » 2 2¼Cosine, limited to 0, S @§ S S·Tangent, limited to , 2 2¹On these restricted domains, we can define the inverse sine, inverse cosine, and inversetangent functions.Inverse Sine, Cosine, and Tangent Functionsª S SºFor angles in the interval « , » , if sin T a , then sin 1 a T 2 2¼For angles in the interval 0, S @ , if cos T a , then cos 1 a T§ S S·For angles in the interval , , if tan T 2 2¹a , then tan 1 aª S Sºsin 1 x has domain [-1, 1] and range « , » 2 2¼ 1cos x has domain [-1, 1] and range 0, S @§ S S·tan 1 x has domain of all real numbers and range , 2 2¹T

Section 6.3 Inverse Trig Functions 383sin 2 T cos 2 T1Using our known value for cosine2§4·sin 2 T 1 5¹16sin 2 T 1 2593sin T rr255Solving for sineSince we know that the inverse cosine always gives an angle on the interval 0, S @ , we§§ 4 ··know that the sine of that angle must be positive, so sin cos 1 sin(T ) 5 ¹¹ 35Example 6§§ 7 ··Find an exact value for sin tan 1 . 4 ¹¹ While we could use a similar technique as in the last example, wewill demonstrate a different technique here. From the inside, we7know there is an angle so tan T. We can envision this as the4opposite and adjacent sides on a right triangle.Using the Pythagorean Theorem, we can find the hypotenuse ofthis triangle:4 2 7 2 hypotenuse 2hypotenuse7ș465Now, we can evaluate the sine of the angle as opposite side divided by hypotenuse7sin T65This gives us our desired composition§7§ 7 ··.sin tan 1 sin(T )65 4 ¹¹ Try it Now§§ 7 ··4. Evaluate cos sin 1 . 9 ¹¹

384 Chapter 6We can also find compositions involving algebraic expressions.Example 7§§ x ··Find a simplified expression for cos sin 1 , for 3 d x d 3 . 3 ¹¹ x. Using the Pythagorean Theorem,3Using our known expression for sineWe know there is an angle ș so that sin Tsin 2 T cos 2 T12§ x·2 cos T 13 ¹x22cos T 1 9cos Tr9 x29Solving for cosiner9 x23ª S SºSince we know that the inverse sine must give an angle on the interval « , » , we 2 2¼can deduce that the cosine of that angle must be positive. This gives us§§ x ··cos sin 1 3 ¹¹ 9 x23Try it Now5. Find a simplified expression for sin tan 1 4 x , for 11dxd .44Important Topics of This SectionInverse trig functions: arcsine, arccosine and arctangentDomain restrictionsEvaluating inverses using unit circle values and the calculatorSimplifying numerical expressions involving the inverse trig functionsSimplifying algebraic expressions involving the inverse trig functions

Section 6.3 Inverse Trig Functions 385Try it Now Answers1. a) S2b) S42. 1.9823 or 113.578 3S44 24.94x5.16 x 2 13.c) Sd)S3

386 Chapter 6Section 6.3 ExercisesEvaluate the following expressions.§ 2·§ 3·1. sin 1 2. sin 1 2 ¹ 2 ¹§ 2·§1·5. cos 1 6. cos 1 2¹ 2 ¹10. tan 19. tan 1 13§2·7. cos 1 2 ¹§4. sin 1 §8. cos 1 11. tan 1 312. tan 1 1§ 1·3. sin 1 2¹Use your calculator to evaluate each expression.14. cos 1 0.815. sin 1 0.813. cos 1 0.42· 2 ¹3· 2 ¹16. tan 1 6Find the angle ș.10ș717.Evaluate the following expressions.§ § S ··19. sin 1 cos 4 ¹¹§ § 4S · ·21. sin 1 cos 3 ¹¹§§ 3 ··23. cos sin 1 7 ¹¹ 25. cos tan 1 4ș18.1219§ § S ··20. cos 1 sin 6 ¹¹§ § 5S · ·22. cos 1 sin 4 ¹¹§§ 4 ··24. sin cos 1 9 ¹¹ §§ 1 ··26. tan sin 1 3 ¹¹ Find a simplified expression for each of the following.§§§ x ··§ x ··27. sin cos 1 , for 5 d x d 528. tan cos 1 , for 2 d x d 2 2 ¹¹ 5 ¹¹ 29. sin tan 1 3 x30. cos tan 1 4 x

Section 6.3 Inverse Trig Functions 379 Section 6.3 Inverse Trig Functions . In previous sections we have evaluated the trigonometric functions at various angles, but at times we need to know what angle would yield a specific sine, cosine, or tangent value. For this, we need inverse functions. Recall that for a one-to-one function, if . f (a) b,