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## High school geometry

### Unit 5: Lesson 7

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?
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

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.
\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.
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.
degrees

Problem 2
Given triangle, D, E, F, find m, angle, E.
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.
degrees

Problem 3
Given triangle, L, Y, N, find m, angle, Y.
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.
degrees

Challenge problem
Solve the triangle completely. That is, find all unknown sides and unknown angles.