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High school geometry

Unit 5: Lesson 7

Solving for an angle in a right triangle using the trigonometric ratios
Math
High school geometry
Right triangles & trigonometry
Solving for an angle in a right triangle using the trigonometric ratios
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Intro to inverse trig functions

CCSS.Math:
HSG.SRT.C.8
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 angle, L, we know the lengths of the opposite and adjacent sides, so we can write:
tangent, left parenthesis, L, 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, equals, start fraction, 35, divided by, 65, end fraction
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 ratiosInverse 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 fractionright arrowsine, 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 fractionright arrowcosine, 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 fractionright arrowtangent, 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.
FunctionGraph
sine, left parenthesis, x, right parenthesis
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.
sine, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis (also called \arcsin, left parenthesis, x, right parenthesis) |
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.
start fraction, 1, divided by, sine, x, end fraction (also called \csc, left parenthesis, x, right parenthesis) |
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.\begin{aligned} { m\angle L}&=\tan^{-1} \left(\dfrac{\blueD{\text{ opposite }} }{\maroonC{\text{ adjacent}}}\right)&{\gray{\text{Define.}}} \\\\ m\angle L&=\tan^{-1}\left(\dfrac{\blueD{35}}{\maroonC{65}}\right)&{\gray{\text{Substitute values.}}} \\\\ m\angle L &\approx 28.30^\circ &{\gray{\text{Evaluate with a calculator.}}}\end{aligned}

Now let's try some practice problems.

Problem 1
Given triangle, K, I, P, find m, angle, 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, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degrees

Problem 2
Given triangle, D, E, F, find m, angle, 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, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degrees

Problem 3
Given triangle, L, Y, N, find m, angle, 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, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degrees

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.
O, E, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
m, angle, O, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degrees
m, angle, Z, equals
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3, slash, 5
  • a simplified improper fraction, like 7, slash, 4
  • a mixed number, like 1, space, 3, slash, 4
  • an exact decimal, like 0, point, 75
  • a multiple of pi, like 12, space, start text, p, i, end text or 2, slash, 3, space, start text, p, i, end text
degrees

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  • piceratops ultimate style avatar for user Danilo Souza Morães
    Posted 7 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...
    (60 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user 陆 鹏
    Posted 5 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 7 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)?
    (1 vote)
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  • blobby green style avatar for user aaron.stinehour
    Posted 3 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
    (3 votes)
    Default Khan Academy avatar avatar for user
  • male robot hal style avatar for user Joseph Arcila
    Posted 3 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?
    (8 votes)
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  • piceratops tree style avatar for user MorganHinz
    Posted 3 years ago. Direct link to MorganHinz's post “How does the fraction tur...”
    How does the fraction turn into an angle measure?
    (3 votes)
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    • mr pink green style avatar for user David Severin
      Posted 3 years ago. Direct link to David Severin's post “It is not the fraction al...”
      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)
  • mr pink orange style avatar for user Meghna Pradhan
    Posted 6 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.
    (4 votes)
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    • piceratops ultimate style avatar for user Homeskool99
      Posted 6 years ago. Direct link to Homeskool99's post “"To the nearest hundredth...”
      "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)
  • blobby green style avatar for user bayli.sarra
    Posted 5 years ago. Direct link to bayli.sarra's post “How would you plug this i...”
    How would you plug this in a calculator?
    (3 votes)
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    • male robot hal style avatar for user Edmund Tran
      Posted 5 years ago. Direct link to Edmund Tran's post “For problem 1, you have s...”
      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)
  • mr pants teal style avatar for user Swayam Bhoi
    Posted 7 years ago. Direct link to Swayam Bhoi's post “if there is no way we can...”
    if there is no way we can find the inverse functions on paper, then how did the values come up for them
    (1 vote)
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    • leaf red style avatar for user JiM
      Posted 7 years ago. Direct link to JiM's post “The values can be determi...”
      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)
  • winston baby style avatar for user ROSIE
    Posted 2 years ago. Direct link to ROSIE's post “Do you know any good site...”
    Do you know any good sites for calculators? I don't have a calculator that can solve inverses.
    (1 vote)
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