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## Trigonometry

### Unit 1: Lesson 4

Solving for an angle in a right triangle using the trigonometric ratios# Intro to inverse trig functions

CCSS.Math:

Learn about arcsine, arccosine, and arctangent, and how they can be used to solve for a missing angle in right triangles.

Let's take a look at a new type of trigonometry problem. Interestingly, these problems can't be solved with sine, cosine, or tangent.

**A problem:**In the triangle below, what is the measure of angle L?

**What we know:**Relative to angle, L, we know the lengths of the opposite and adjacent sides, so we can write:

But this doesn't help us find the measure of angle, L. We're stuck!

**What we need:**We need new mathematical tools to solve problems like these. Our old friends sine, cosine, and tangent aren’t up to the task. They take angles and give side ratios, but we need functions that take side ratios and give angles. We need

**inverse trig functions**!

## The inverse trigonometric functions

We already know about

*inverse operations*. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. Each operation does the*opposite*of its inverse.The idea is the same in trigonometry.

*Inverse trig functions*do the opposite of the “regular” trig functions. For example:- Inverse sine left parenthesis, sine, start superscript, minus, 1, end superscript, right parenthesis does the opposite of the sine.
- Inverse cosine left parenthesis, cosine, start superscript, minus, 1, end superscript, right parenthesis does the opposite of the cosine.
- Inverse tangent left parenthesis, tangent, start superscript, minus, 1, end superscript, right parenthesis does the opposite of the tangent.

In general, if you know the trig ratio but not the angle, you can use the corresponding inverse trig function to find the angle. This is expressed mathematically in the statements below.

Trigonometric functions input angles and output side ratios | Inverse trigonometric functions input side ratios and output angles | |
---|---|---|

sine, left parenthesis, theta, right parenthesis, equals, start fraction, start text, o, p, p, o, s, i, t, e, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, end text, end fraction | right arrow | sine, start superscript, minus, 1, end superscript, left parenthesis, start fraction, start text, o, p, p, o, s, i, t, e, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, end text, end fraction, right parenthesis, equals, theta |

cosine, left parenthesis, theta, right parenthesis, equals, start fraction, start text, a, d, j, a, c, e, n, t, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, end text, end fraction | right arrow | cosine, start superscript, minus, 1, end superscript, left parenthesis, start fraction, start text, a, d, j, a, c, e, n, t, end text, divided by, start text, h, y, p, o, t, e, n, u, s, e, end text, end fraction, right parenthesis, equals, theta |

tangent, left parenthesis, theta, right parenthesis, equals, start fraction, start text, o, p, p, o, s, i, t, e, end text, divided by, start text, a, d, j, a, c, e, n, t, end text, end fraction | right arrow | tangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, start text, o, p, p, o, s, i, t, e, end text, divided by, start text, a, d, j, a, c, e, n, t, end text, end fraction, right parenthesis, equals, theta |

## Misconception alert!

The expression sine, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis is not the same as start fraction, 1, divided by, sine, left parenthesis, x, right parenthesis, end fraction. In other words, the minus, 1 is not an exponent. Instead, it simply means inverse function.

Function | Graph |
---|---|

sine, left parenthesis, x, right parenthesis |

sine, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis (also called \arcsin, left parenthesis, x, right parenthesis) |

start fraction, 1, divided by, sine, x, end fraction (also called \csc, left parenthesis, x, right parenthesis) |

However, there is an alternate notation that avoids this pitfall! We can also express the inverse sine as \arcsin, the inverse cosine as \arccos, and the inverse tangent as \arctan. This notation is common in computer programming languages, and less common in mathematics.

## Solving the introductory problem

In the introductory problem, we were given the opposite and adjacent side lengths, so we can use inverse tangent to find the angle.

## Now let's try some practice problems.

## Want to join the conversation?

- this might sound like a silly question, but i was hoping that sin(90) = 2 sin(45).

Why doesn't that work? Trig functions are all about ratios and relations, the least i could expect was to find a relation like that...(3 votes)- this might have been possible if sin was a linear function which its not....(32 votes)

- How to calculate the inverse function in a calculator?(9 votes)
- Many calculators (TI and others) have the inverse trig funcdtions (sin-1, cos-1, tan-1) on the same button, but using the 2nd sin function. Do not know which particular calculator you are talking about.(3 votes)

- So I know that arcsin ( sin(x) ) = x but... what happens when you do arcsin(x) * sin(x)?(1 vote)
- It would be the same thing as multiplying the angle by the two side ratio(3 votes)

- when I do inverse sin(10/8) I get an error. I used mutable calculators and they all give errors(3 votes)
- sin(x) only returns numbers between 0 and 1, so arcsin(x) can only accept numbers between 0 and 1. 10/8 is greater than 1, so you're trying to input a number outside the domain.(12 votes)

- How does the fraction turn into an angle measure?(3 votes)
- It is not the fraction alone, but the inverse function of the fraction. The idea what you have learned in Algebra, we move things to isolate the variable by opposites. You know add/subtract and multiply/divide are opposites, you may know squares and square roots are opposites, so this is the opposites for trig functions. If you have the sin(X)=4/5, the opposite operation of sin is sin-1. so sin-1(sin(x))=sin-1 (4/5), this is based on if you do something to one side, you do the same to the other. Then if you do it right, something cancels (so sin-1(sin) cancels just as any other opposites and you are left with x = sin-1 (4/5).(5 votes)

- could some one explain what ' round your answer to the nearest hundredth degree' means. its mentioned in the second practice question.(4 votes)
- "To the nearest hundredth of a degree" means to solve it, and then round it to 2 decimal places. The first place is tenths, and the second place is hundredths.

Example: Problem 3.

We're trying to find angle Y. We have the adjacent side length and the hypotenuse length. With the sides adjacent and hypotenuse, we can use the Cosine function to determine angle Y.

CosY = adj/hyp

CosY = 3/10

CosY = 0.30

This is where the Inverse Functions come in. If we know that CosY = 0.30, we're trying to find the angle Y that has a Cosine 0.30. To do so:

-Enter 0.30 on your calculator

-Find the Inverse button, then the Cosine button (This could also be the Second Function button, or the Arccosine button).

Should come out to 72.542397, rounded.

To round to the nearest hundredth of a degree, we round to 2 decimal, places, giving the answer 72.54.(2 votes)

- How would you plug this in a calculator?(3 votes)
- For problem 1, you have sin(m∠I) = 8/10 (opposite over hypotenuse). So that means the measure of angle I is the inverse sine of (8/10). On most calculators, the inverse trig functions are the secondary functions of the trig functions. So on a TI-84, for example, you would press "2nd" and then "SIN" to do inverse sine. Next, put 8/10, close the parentheses, and press "Enter".(4 votes)

- if there is no way we can find the inverse functions on paper, then how did the values come up for them(1 vote)
- The values can be determined, (to good approximation), by using something called a power series. A power series of a function is a polynomial with infinitely many terms that is exactly equal to the function over some region, (you learn about these in calculus, and they're one of the most important things in a scientist's tool belt). If you calculate enough of the terms in the power series expansion of a function, then you can calculate the value of the function to arbitrary precision.(5 votes)

- I can't find a calculator that will let me perform these functions or even calculate them because I need the specific symbols for the trigonometric functions. Can Khan Academy please include and provide a calculator that displays and allows these symbols that we have to use in order to get the correct answers.(3 votes)
- You can find a scientific calculator on all problems that require them. Alternatively you could use google to calculate it on their free calculator.(2 votes)

- This doesn't make sense for me. In the first question it says put

tan−1 (35 divided by 65) = 28.30*

But when I put tan-1 (35/65) into my calculator it came up with .493941

(I used the inverse function, not -1, my computer just shows it like a minus one.)

I know I'm missing something here and I would like to know so I can move on. I tried putting in the other equations too but they all got different answers which weren't a degree answer or the number given.

(Edit, read reply below)(2 votes)- Nvm, I was able to find out the problem. Who anyone else who has the same problem you need to change it from RAD to DEG. Took me a few minutes to figure that out.(4 votes)