1. 7-6 The Inverse Trigonometric Functions
Objective: To find values of the inverse trigonometric functions.

2. The Inverse Trigonometric Function
When does a function have an inverse?
It means that the function is one-to-one.
One-to-one means that every x-value is assigned no more than one y-value AND every y-value is assigned no more than one x-value.
How do you determine if a function has an inverse?
Use the horizontal line test (HLT).

3. The Inverse Trigonometric Function
Inverse Sine Function
Recall that for a function to have an inverse, it must be a one-to-one function and pass the Horizontal Line Test.
f(x) = sin x does not pass the Horizontal Line Test
and must be restricted to find its inverse. y x
y = sin x
–/2
/2
sin x has an inverse function on this interval.

4. The Inverse Trigonometric Function
The inverse sine function is defined by
y = arcsin x if and only if sin y = x.
Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians).
Angle whose sine is x
The domain of y = arcsin x is [–1, 1].
The range of y = arcsin x is [–/2 , /2].
Example 1:
This is another way to write arcsin x.

5. The Graph of Inverse Sine

6. The Inverse Sine Function
The inverse sine function, denoted by sin-1, is the inverse of the restricted sine function y = sin x, -  /2 < x <  / 2. Thus,
y = sin-1 x means sin y = x,
where -  /2 < y <  /2 and –1 < x < 1. We read y = sin-1 x as “ y equals the inverse sine at x.” y -1 1
 /2 x
-  /2
y = sin x
-  /2 < x < /2
Domain: [-  /2,  /2]
Range: [-1, 1]

7. Finding Exact Values of sin-1x
Let  = sin-1 x.
Rewrite step 1 as sin  = x.
Use the exact values in the table to find the value of  in [-/2 , /2] that satisfies sin  = x.

8. Example
Find the exact value of sin-1(1/2)

9. The Inverse Trigonometric Function
The other inverse trig functions are generated by using similar restrictions on the domain of the trig function. Consider the cosine function:
Inverse Cosine Function
f(x) = cos x must be restricted to find its inverse. y x
y = cos x 
cos x has an inverse function on this interval.

10. The Inverse Trigonometric Function
The inverse cosine function is defined by
y = arccos x if and only if cos y = x.
Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians).
Angle whose cosine is x
The domain of y = arccos x is [–1, 1].
The range of y = arccos x is [0 , ].
Example 2:
This is another way to write arccos x.

11. The Graph of Inverse Cosine

12. The Graph of Inverse Cosine
What is the relation between arcsin(x) and arccos(x) ?
arccos(x) = (-1)arcsin(x) + /2
arcsin(x) + arccos(x) = /2

13. The Inverse Cosine Function
The inverse cosine function, denoted by cos-1, is the inverse of the restricted cosine function
y = cos x, 0 < x < . Thus,
y = cos-1 x means cos y = x,
where 0 < y <  and –1 < x < 1.

14. Inverses of Sine and Cosine
Sin(x)
Domain:
Range:
-1≤y≤1
Arccos(x)
Domain:
-1≤x≤1
Range:
0≤y≤¶
Cos(x)
Domain:
0≤x≤¶
Range:
-1≤y≤1
Arcsin(x)
Domain:
-1≤x≤1
Range: 2 p
< - y

15. Example Find the exact value of cos-1 (-3 /2) Solution
Step 1 Let  = cos-1 x. Thus =cos-1 (-3 /2)
We must find the angle , 0 <  < , whose cosine equals -3 /2
Step 2 Rewrite  = cos-1 x as cos  = x. We obtain cos  = (-3 /2)

16. Example cont. Find the exact value of cos-1 (-3 /2) Solution
Step 3 Use the exact values in the table to find the value of  in [0, ] that satisfies cos  = x. The table on the previous slide shows that the only angle in the interval [0, ] that satisfies cos  = (-3 /2) is 5/6. Thus,  = 5/6

17. The Inverse Trigonometric Function
The other trig functions require similar restrictions on their domains in order to generate an inverse.
Like the sine function, the domain of the section of the
tangent that generates the arctan is

18. The Inverse Trigonometric Function
The inverse tangent function is defined by
y = arctan x if and only if tan y = x.
Unless you are instructed to use degrees, you should assume that inverse trig functions will generate outputs of real numbers (in radians).
Angle whose tangent is x
The domain of y = arctan x is (-,) .
The range of y = arctan x is (–/2 , /2).
Example 3:
This is another way to write arctan x.

19. The Inverse Tangent Function
The inverse tangent function, denoted by tan-1, is the inverse of the restricted tangent function
y = tan x, -/2 < x < /2. Thus,
y = tan-1 x means tan y = x,
where -  /2 < y <  /2 and –  < x < .

20. Graph of Tangent of x with an unrestricted domain.

21. Inverse Tangent of x Domain: Range: Real Numbers-1
Tan x = y means that tan y = x and 2 p
< - y

22. The Graph of Inverse Tangent and Cotangent
What is the relation between arctan(x) and arcot(x) ?
arccot(x) = (-1)arctan(x) + /2
arctan(x) + arccot(x) = /2

23. Tan can also be called arctan-1

24. Composition of Functions
f(f –1(x)) = x and (f –1(f(x)) = x.
Inverse Properties:
If –1  x  1 and – /2  y  /2, then
sin(arcsin x) = x and arcsin(sin y) = y.
If –1  x  1 and 0  y  , then
cos(arccos x) = x and arccos(cos y) = y.
If x is a real number and –/2 < y < /2, then
tan(arctan x) = x and arctan(tan y) = y.
If x is a real number and 0 < y < , then
cot(arccot x) = x and arccot(cot y) = y.
Example 5: tan(arctan 4) = 4
Composition of Functions

25. Inverse Properties The Sine Function and Its Inverse
sin (sin-1 x) = x for every x in the interval [-1, 1].
sin-1(sin x) = x for every x in the interval [-/2,/2].
The Cosine Function and Its Inverse
cos (cos-1 x) = x for every x in the interval [-1, 1].
cos-1(cos x) = x for every x in the interval [0, ].
The Tangent Function and Its Inverse
tan (tan-1 x) = x for every real number x
tan-1(tan x) = x for every x in the interval (-/2,/2).

26. Inverse Trigonometric Functions
Example: sin(arcsin(-1)) = -1 arcsin(sin(pi/4)) = pi/4, (domain between -pi/2 and pi/2). Another example: If you have sin(x) = 1 if you take arcsin of both sides: arcsin(sin(x)) =arcsin( 1) You get: x = arcsin(1) = pi/2 In another word, "what x can I choose so that if I take its sine, I will get 1"
Consider tan(x) with domain
It’s inverse is arctan(x) or tan (x)
The relationship between them is:
tan(tan (x)) = x
tan (tan(x)) = x -1 -1 -1
This is the same for other trigonometric functions.

27. Inverse Keys are on your calculator.

29. The Inverse Function of Sine
Basic idea: To find sin^-1(½), we ask "what angle has sine equal to ½?" The answer is 30°. As a result we say sin^-1(½) = 30°. In radians this is sin-1(½) = π/6.
More: There are actually many angles that have sine equal to ½. We are really asking "what is the simplest, most basic angle that has sine equal to ½?" As before, the answer is 30°. Thus sin^-1(½) = 30° or sin^-1(½) = π/6.
Details: What is sin^-1(–½)? Do we choose 210°, –30°, 330° , or some other angle? The answer is –30°. With inverse sine, we select the angle on the right half of the unit circle having measure as close to zero as possible. Thus sin-1(–½) = –30° or sin–1(–½) = –π/6.
In other words, the range of sin-1 is restricted to [–90°, 90°] or .
Note: arcsin refers to "arc sine", or the radian measure of the arc on a circle corresponding to a given value of sine.
Technical note: Since none of the six trig functions sine, cosine, tangent, cosecant, secant, and cotangent are one-to-one, their inverses are not functions. Each trig function can have its domain restricted, however, in order to make its inverse a function.

30. Question??
If you had sin(x) / sin²(x), you are left with 1/sin(x), right? So how come when you have sin(x)•arcsin(x) you aren't left with 1? Isn't arcsin(x) the same as writing sin (x)? Therefore writing sin(x)arcsin(x) would be like writing sin(x)/sin(x) = 1 But it isn't.... why not? -1

31. sin(arcsin(x)) = arcsin(sin(x)) = x
Answer:
Because sin is not 1/sin(x). The reciprocal of the sine function is the cosecant function.
The arcsine function is the inverse function for the sine function on the interval So they “cancel” each other under the composition of functions.
sin(arcsin(x)) = arcsin(sin(x)) = x
The notation for inverse functions in just a shorthand way of writing the inverse. The -1 looks like an exponent but it is not an exponent.

32. Evaluate each expression without using a calculator.
If = x, the sinx = -1 and
Since sin (-π/2) = -1, then
Sin (-1) = (π/2) 2 p
< - x
Whose tangent is √3
Since tan π/3 = √3 then
Tan √3 = π/3

33. Find using the degree and radian mode on the calculator
150˚

34. Find the following: Find Sin (0.8) with a calculator.-1
Find Sin (0.8) with a calculator.
Degree mode = 53˚
Radian mode = 0.93
Find Cos (-0.5) without a calculator.
Cos (-0.5) = x means that cos x =-0.5 between 0 and π. Thus,
Cos (-0.5) = 2π/3 -1 -1 -1

35. Find with and without a calculator.-2 3
Hypotenuse² will be (-2)² + 3² = √13
The cos is adj/hyp = 3/√13
Rationalize Denominator = 3√13/13
√13
Calculator answer ≈ 0.83

36. Find the approximate value (calculator) and exact value (without a calculator)
csc(cos (-0.4)) -1 5 -2
-0.4 in fraction form is -2/5
Cos = adj/hyp
Opp. =√ 5² - (-2)² = √21
Csc = 1/sin = hyp/opp = 5/√21
Rationalize denominator = 5√21/21
Calculator: 1.09

37. Example 8: a. sin–1(sin (–/2)) = –/2
does not lie in the range of the arcsine function, –/2  y  /2. y x
However, it is coterminal with which does lie in the range of the arcsine function.

38. Example 9: a. sin–1(sin (–3/2)) = /2
does not lie in the range of the arcsin function, –/2  y  /2. y x
However, it is coterminal with which does lie in the range of the arcsin function.

39. Example 10:
[Solution] x y 3 u 2

40. Finally, we encounter the composition of trig functions with inverse trig functions. The following are pretty straightforward compositions. Try them yourself before you click to the answer. so
First, what do we know about
We know that is an angle whose sine is
Did you suspect from the beginning that this was the answer because that is the way inverse functions are SUPPOSED to behave? If so, good instincts but….

41. Consider a slightly different setup:
This is also the composition of two inverse functions but…
Did you suspect the answer was going to be 2/3? This problem behaved differently because the first angle, 2/3, was outside the range of the arcsin. So use some caution when evaluating the composition of inverse trig functions.
The remainder of this presentation consists of practice problems, their answers and a few complete solutions.

42. Find the exact value of each expression without using a calculator
Find the exact value of each expression without using a calculator. When your answer is an angle, express it in radians. Work out the answers yourself before you click.

43. On most calculators, you access the inverse trig functions by using the 2nd function option on the corresponding trig functions. The mode button allows you to choose whether your work will be in degrees or in radians. You have to stay on top of this because the answer is not in a format that tells you which mode you are in.
Use a calculator. For 21-24, express your answers in radians rounded to the nearest hundredth.
Use a calculator. For 17-20, round to the nearest tenth of a degree.

44. Use a calculator. When your answer is an angle, express it in radians rounded to the hundredth’s place. When your answer is a ratio, round it to four decimal places, but don’t round off until the very end of the problem.
Answers appear in the following slides.

45. Answers for problems 1 – 9.
Negative ratios for arccos generate angles in Quadrant II. y x 1 2
The reference angle is
so the answer is

46. y x -1 2
14. x 1 2 y
15.

47. Answers for 17 – 30.

48. Assignment
Page 289 #2, 4, 5 – 8, 11 – 14
Chapter 7 Test Wednesday