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## High school geometry

### Course: High school geometry > Unit 5

Lesson 7: 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...(70 votes)- this might have been possible if sin was a linear function which its not....(48 votes)

- Love the site, but slightly thrown having to switch from using DEG mode to RAD mode to get correct answer on inverse trig questions. Would be good to be given a heads-up that this was necessary. And why it was necessary. Which I. Still haven't really figured out!(22 votes)
- DEG mode stands for "degree". This means that your calculator interprets and outputs angles in the unit of degrees. RAD mode stands for "radian". This means that your calculator interprets and outputs angles in the unit of radians. If you are not sure what radians are, I suggest you watch the KA videos on them. Switching between DEG mode and RAD mode on a calculator is similar to switching between "miles per hour" and "kilometers per hour" on a speedometer. You still get the same speeds, but in different units.

Comment if you have questions!(54 votes)

- how to turn calculator on(17 votes)
- If it is not turning on then you need to replace the batteries. Hope this helps.

(:(22 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.(21 votes)

- So I know that arcsin ( sin(x) ) = x but... what happens when you do arcsin(x) * sin(x)?(6 votes)
- It would be the same thing as multiplying the angle by the two side ratio(4 votes)

- What if we do not want to use a calculator and do it manually?(4 votes)
- Then you will need access to trigonometric tables that you can read in reverse. This is how I used to estimate the inverse trigonometric functions when I was in high school. I still have a book of tables to trig functions, logarithms, and z-scores (among other useful relationships) to which I refer when solving some problems, but the modern method of using a calculator or computer to access this information is usually more efficient and precise.(17 votes)

- could some one explain what ' round your answer to the nearest hundredth degree' means. its mentioned in the second practice question.(3 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.(17 votes)

- What happens behind the scene when I compute in the calculator arcsin (arcos or arctan) of some number. is it possible to calculate it without a calculator? what are their functions?(10 votes)
- when I do inverse sin(10/8) I get an error. I used mutable calculators and they all give errors(1 vote)
- 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.(14 votes)

- if there is no way we can find the inverse functions on paper, then how did the values come up for them(4 votes)
- 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.(6 votes)