If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Trigonometry

Course: Trigonometry > Unit 1

Lesson 4: Solving for an angle in a right triangle using the trigonometric ratios
Math
Trigonometry
Right triangles & trigonometry
Solving for an angle in a right triangle using the trigonometric ratios
© 2023 Khan AcademyTerms of usePrivacy PolicyCookie Notice

Intro to inverse trig functions

Google Classroom
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?
A right triangle with leg lengths of thirty-five and sixty-five. Angle L is opposite the short leg.and is unknown.
What we know: Relative to ∠L, we know the lengths of the opposite and adjacent sides, so we can write:
tan⁡(L)=oppositeadjacent=3565
But this doesn't help us find the measure of ∠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 (sin−1) does the opposite of the sine.
  • Inverse cosine (cos−1) does the opposite of the cosine.
  • Inverse tangent (tan−1) 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 ratiosInverse trigonometric functions input side ratios and output angles
sin⁡(θ)=oppositehypotenuse→sin−1⁡(oppositehypotenuse)=θ
cos⁡(θ)=adjacenthypotenuse→cos−1⁡(adjacenthypotenuse)=θ
tan⁡(θ)=oppositeadjacent→tan−1⁡(oppositeadjacent)=θ

Misconception alert!

The expression sin−1⁡(x) is not the same as 1sin⁡(x). In other words, the −1 is not an exponent. Instead, it simply means inverse function.
FunctionGraph
sin⁡(x)
A coordinate plane. The x-axis starts at zero and goes to ninety by tens. It is labeled degrees. The y-axis starts at zero and goes to two by two tenths. It is labeled a ratio. The graphed line is labeled sine of x, which is a nonlinear curve. The line for the sine of x starts at the origin and passes through the points twenty-four, zero point four, forty, zero point sixty-seven, fifty-two, zero point eight, and ninety, one. It is increasing from the origin to the point ninety, one. The rate of change gets smaller, or shallower, as the degrees, or x-values, get larger. All points are approximations.
sin−1⁡(x) (also called arcsin⁡(x)) |
A coordinate plane. The x-axis starts at zero and goes to two by two tenths. It is labeled a ratio. The y-axis starts at zero and goes to ninety by tens. It is labeled degrees. The graphed line is labeled inverse sine of x, which is a nonlinear curve. The line for the inverse sine of x starts at the origin and passes through the points zero point four, twenty-four, zero point sixty-seven, forty, zero point eight, fifty-two, and one, ninety. It is increasing from the origin to the point one, ninety. The rate of change gets larger, or sharper, as the ratios, or x-values, get larger. All points are approximations.
1sin⁡x (also called csc⁡(x)) |
A coordinate plane. The x-axis starts at zero and goes to ninety by tens. It is labeled degrees. The y-axis starts at zero and goes to two by two tenths. It is labeled a ratio. The graphed line is one divided by the sine of x, which is a nonlinear curve. The line for the cosecant of x starts by decreasing from the point thirty, two. It continues decreasing until the point ninety, one. The rate of change starts steep at the point thirty, two, but it get smaller at the graph goes through the points forty, one point fifty-five, fifty, one point three, and sixty-five, one point one. The rate of change is very shallow as the graph approaches the point ninety, one. All points are approximations.
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.
A right triangle with vertices L and V where angle L is unknown. The side between angles L and ninety degrees is sixty-five degress. The side between the right angle and the vertex V is thirty-five units.
m∠L=tan−1⁡( opposite  adjacent)Define.m∠L=tan−1⁡(3565)Substitute values.m∠L≈28.30∘Evaluate with a calculator.

Now let's try some practice problems.

Problem 1
Given △KIP, find m∠I.
Round your answer to the nearest hundredth of a degree.
Right triangle K I P where angle A P I is a right angle. Angle K I P is an unknown angle. K I is ten units. K P is eight units.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
∘

Problem 2
Given △DEF, find m∠E.
Round your answer to the nearest hundredth of a degree.
Right triangle D E F where angle D F E is a right angle. Angle D E F is an unknown angle. D F is four units. E F is six units.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
∘

Problem 3
Given △LYN, find m∠Y.
Round your answer to the nearest hundredth of a degree.
Right triangle L Y N where angle Y L N is a right angle. Angle L Y N is an unknown angle. Y N is ten units. L Y is three units.
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
∘

Challenge problem
Solve the triangle completely. That is, find all unknown sides and unknown angles.
Round your answers to the nearest hundredth.
Right Triangle O Z E where angle O E Z is a right angle. Side O Z is nine units. Side E Z is four units.
OE=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
m∠O=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
∘
m∠Z=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
∘

Want to join the conversation?

Log in
  • piceratops ultimate style avatar for user Danilo Souza Morães
    Posted 8 years ago. Direct link to Danilo Souza Morães's post “this might sound like a s...”
    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...
    (71 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user mickbeckley
    Posted 6 years ago. Direct link to mickbeckley's post “Love the site, but slight...”
    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)
    Default Khan Academy avatar avatar for user
    • mr pink red style avatar for user andrewp18
      Posted 6 years ago. Direct link to andrewp18's post “DEG mode stands for "degr...”
      Incredible Answer
      Great Answer
      Good Answer
      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!
      (56 votes)
  • blobby green style avatar for user Margaret DeCoursey
    Posted 3 years ago. Direct link to Margaret DeCoursey's post “how to turn calculator on”
    how to turn calculator on
    (19 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user 陆 鹏
    Posted 6 years ago. Direct link to 陆 鹏's post “How to calculate the inve...”
    How to calculate the inverse function in a calculator?
    (9 votes)
    Default Khan Academy avatar avatar for user
  • winston default style avatar for user Jinho Yoon
    Posted 8 years ago. Direct link to Jinho Yoon's post “So I know that arcsin ( s...”
    So I know that arcsin ( sin(x) ) = x but... what happens when you do arcsin(x) * sin(x)?
    (6 votes)
    Default Khan Academy avatar avatar for user
  • female robot grace style avatar for user Kritika Patnaik
    Posted 5 years ago. Direct link to Kritika Patnaik's post “What if we do not want to...”
    What if we do not want to use a calculator and do it manually?
    (4 votes)
    Default Khan Academy avatar avatar for user
    • starky ultimate style avatar for user Paul Miller
      Posted 5 years ago. Direct link to Paul Miller's post “Then you will need access...”
      Good Answer
      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)
  • blobby green style avatar for user mj2026
    Posted 5 months ago. Direct link to mj2026's post “Every time I use one of t...”
    Every time I use one of the formulas, I get a much smaller answer than I’m supposed to. Even in the introductory problem, if I put tan^-1(35/65) into my calculator, I get the answer of ~0.49 when its supposed to be ~28.30. I can’t find a way to get the correct answers, am I missing a step?
    (5 votes)
    Default Khan Academy avatar avatar for user
  • mr pink orange style avatar for user Meghna Pradhan
    Posted 7 years ago. Direct link to Meghna Pradhan's post “could some one explain wh...”
    could some one explain what ' round your answer to the nearest hundredth degree' means. its mentioned in the second practice question.
    (3 votes)
    Default Khan Academy avatar avatar for user
    • piceratops ultimate style avatar for user Homeskool99
      Posted 7 years ago. Direct link to Homeskool99's post “"To the nearest hundredth...”
      Good Answer
      "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)
  • male robot hal style avatar for user Joseph Arcila
    Posted 4 years ago. Direct link to Joseph Arcila's post “What happens behind the s...”
    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)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user aaron.stinehour
    Posted 4 years ago. Direct link to aaron.stinehour's post “when I do inverse sin(10/...”
    when I do inverse sin(10/8) I get an error. I used mutable calculators and they all give errors
    (1 vote)
    Default Khan Academy avatar avatar for user