Solving for an angle in a right triangle using the trigonometric ratios

Intro to inverse trig functions

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:
tangent, left parenthesis, L, right parenthesis, equals, start fraction, o, p, p, o, s, i, t, e, divided by, a, d, j, a, c, e, n, t, 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, o, p, p, o, s, i, t, e, divided by, h, y, p, o, t, e, n, u, s, e, end fractionright arrowsine, start superscript, minus, 1, end superscript, left parenthesis, start fraction, o, p, p, o, s, i, t, e, divided by, h, y, p, o, t, e, n, u, s, e, end fraction, right parenthesis, equals, theta
cosine, left parenthesis, theta, right parenthesis, equals, start fraction, a, d, j, a, c, e, n, t, divided by, h, y, p, o, t, e, n, u, s, e, end fractionright arrowcosine, start superscript, minus, 1, end superscript, left parenthesis, start fraction, a, d, j, a, c, e, n, t, divided by, h, y, p, o, t, e, n, u, s, e, end fraction, right parenthesis, equals, theta
tangent, left parenthesis, theta, right parenthesis, equals, start fraction, o, p, p, o, s, i, t, e, divided by, a, d, j, a, c, e, n, t, end fractionright arrowtangent, start superscript, minus, 1, end superscript, left parenthesis, start fraction, o, p, p, o, s, i, t, e, divided by, a, d, j, a, c, e, n, t, 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.
[I'm confused. Please explain this further.]
If a number or variable is raised to the minus, 1 power, then this refers to the multiplicative inverse, or the reciprocal. For example, 3, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 3, end fraction. In general, if a is a nonzero real number, then a, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, a, end fraction.
However, this is not the case for sine, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis. This is because the sine is a function, not a quantity!
In general, whenever you see a raised minus, 1 after a function name, it refers to the inverse function. So, for example, if f is a function, then f, start superscript, minus, 1, end superscript represents the inverse of function f. The expression f, start superscript, minus, 1, end superscript, left parenthesis, x, right parenthesis represents the value of the inverse function for the input x.
However, there is an alternate notation that avoids this pitfall! The inverse sine can also be expressed as arcsin\arcsin, the inverse cosine as arccos\arccos, and the inverse tangent as arctan\arctan. This notation is common in computer programming languages, but not 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.
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{\text{} \blueD{\text{ opposite }} }{\text{}\maroonC{\text{ adjacent} }\text{ }} \right)\quad\small{\gray{\text{Define.}}} \\\\ m\angle L&=\tan^{-1}\left(\dfrac{\blueD{35}}{\maroonC{65}}\right)\quad\small{\gray{\text{Substitute values.}}} \\\\ m\angle L &\approx 28.30^\circ \quad\small{\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.
  • 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, p, i or 2, slash, 3, space, p, i
degree

Explain
Problem 2
Given triangle, D, E, F, find m, angle, E.
Round your answer to the nearest hundredth of a degree.
  • 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, p, i or 2, slash, 3, space, p, i
degree

Explain
Problem 3
Given triangle, L, Y, N, find m, angle, Y.
Round your answer to the nearest hundredth of a degree.
  • 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, p, i or 2, slash, 3, space, p, i
degree

Explain
Challenge problem
Solve the triangle completely. That is, find all unknown sides and unknown angles.
Round your answers to the nearest hundredth.
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, p, i or 2, slash, 3, space, p, i
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, p, i or 2, slash, 3, space, p, i
degree
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, p, i or 2, slash, 3, space, p, i
degree

Explain