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Mathematics /AP Calculus AB: 6.5.3 Evaluating Inverse Trigonometric Functions

AP Calculus AB: 6.5.3 Evaluating Inverse Trigonometric Functions

Mathematics12 CardsCreated 8 months ago

This section focuses on evaluating inverse trigonometric expressions by rewriting them as standard trig equations and solving for the angle within the correct restricted domain. It also emphasizes calculator usage, the importance of radian mode in calculus, and understanding that inverse trig functions do not always exactly "undo" the original trig functions unless inputs fall within their defined domains.

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Evaluating Inverse Trigonometric Functions

To evaluate inverse trigonometric expressions, first convert them into standard trig expressions. Use this technique to solve inverse trig equations as well.
An inverse trig function will not reverse the original function outside of the domain of the inverse trig functions.

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Key Terms

Term

Definition

Evaluating Inverse Trigonometric Functions

To evaluate inverse trigonometric expressions, first convert them into standard trig expressions. Use this technique to solve inverse trig ...

note

The key to evaluating inverse trigonometric expressions is to convert them into standard trig expressions.
To find arcsin (1...

Evaluate cos−1(1/2)=y.

π/3

Solve tan−1(3x−1)=π4

2/3

Solve sin−1(2x)=π/6.

1/4

Evaluate y = cot−1 (2).

0.46

Evaluate y=sin−1(sin(2π/3)).

π/3

Evaluate sin−1√3/2=y.

π/3

Evaluate tan−1 (−1) = y.

−π/4

Evaluate y=cos−1(cos(−2π/3)).

2π/3

Solve sec−1(x2+1)=π/3

1 or −1

Evaluate y=tan−1(tan(5π/6))

−π/6

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AP Calculus AB: 6.5.3 Evaluating Inverse Trigonometric Functions

Term	Definition
Evaluating Inverse Trigonometric Functions	To evaluate inverse trigonometric expressions, first convert them into standard trig expressions. Use this technique to solve inverse trig equations as well. An inverse trig function will not reverse the original function outside of the domain of the inverse trig functions.
note	The key to evaluating inverse trigonometric expressions is to convert them into standard trig expressions. To find arcsin (1/2), you need to look for “the angle whose sine is 1/2.” In other words, look for x such that sin x = 1/2. Of course, there are lots of angles for which the sine is 1/2. But only one of them lies in the restricted domain for sine that you used to define arcsine. There are lots of angles for which tangent is one, but only one of them lies in the restricted domain. That’s what makes arctangent a function. Most scientific calculators have buttons for evaluating inverse trigonometric functions. The buttons usually have the raised –1 notation. Before you start, decide whether you want the answer in degrees or radians. Calculus usually prefers radians. To evaluate arctan (1), you will need to press the tan –1 key. For some calculators you will need to enter 1 and then press the tan –1 key. Graphing calculators like Professor Burger’s usually require you to press tan –1 and then 1. Even though the calculator is in radian mode, it usually presents its answer as a decimal. Of course, your calculator may work differently, so check the user’s manual. To evaluate arcsin (sin (π)), you need to evaluate the inside first. Then the expression becomes arcsin (0), which you can transform to look for “the angle whose sine is zero.” So the value of arcsin (sin (π)) is zero. Wait a minute! The inverse function machine is supposed to give back what the original machine started with. The two functions should cancel and produce π! This is why the intervals are important. Arcsine is only defined from –π/2 to π/2. If you start with a number in this interval, then the functions will cancel each other. Remember, if you have a different calculator, you may have to press the buttons in the reverse order. The calculator cannot think, it can only calculate. It may produce inaccurate results if it is not in the correct mode. Make sure to evaluate the expression based on your knowledge of the functions before using the calculator to check. You can solve inverse trig equations using the same method you used to evaluate inverse trig expressions. First think backwards by expressing the equation in terms of standard trig functions. Evaluate the trig expression and solve like any other equation. Check your work by plugging the solution back in to the original equation to verify that it is a solution.
Evaluate cos−1(1/2)=y.	π/3
Solve tan−1(3x−1)=π4	2/3
Solve sin−1(2x)=π/6.	1/4
Evaluate y = cot−1 (2).	0.46
Evaluate y=sin−1(sin(2π/3)).	π/3
Evaluate sin−1√3/2=y.	π/3
Evaluate tan−1 (−1) = y.	−π/4
Evaluate y=cos−1(cos(−2π/3)).	2π/3
Solve sec−1(x2+1)=π/3	1 or −1
Evaluate y=tan−1(tan(5π/6))	−π/6