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# Inverse Trigonometric Ratios

## Solving for an angle given a trigonometric ratio.

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Inverse Trigonometric Ratios

The maximum slope of a wheelchair ramp is 1:12. For a wheelchair ramp made with these specifications, what angle does the ramp make with the flat ground?

### Inverse Trigonometric Ratios

Recall that the sine, cosine, and tangent of angles are ratios of pairs of sides in right triangles.

• The sine of an angle in a right triangle is the ratio of the side opposite the angle to the hypotenuse.
• The cosine of an angle in a right triangle is the ratio of the side adjacent to the angle to the hypotenuse.
• The tangent of an angle in a right triangle is the ratio of the side opposite the angle to the side adjacent to the angle.

You can use the trigonometric ratios to find missing sides of right triangles when given at least one side length and one angle measure. You can use the inverse trigonometric ratios to find a missing angle in a right triangle when given two sides.

• The inverse sine of a ratio gives the angle in a right triangle whose sine is the given ratio. Inverse sine is also called arcsine and is labeled \begin{align*}\sin^{-1}\end{align*} or arcsin.
• The inverse cosine of a ratio gives the angle in a right triangle whose cosine is the given ratio. Inverse cosine is also called arccosine and is labeled \begin{align*}\cos^{-1}\end{align*} or arccos.
• The inverse tangent of a ratio gives the angle in a right triangle whose tangent is the given ratio. Inverse tangent is also called arctangent and is labeled \begin{align*}\tan^{-1}\end{align*} or arctan.

Note that in each case the “-1” is to indicate inverse, and is not an exponent.

To find the measure of an angle using an inverse trigonometric ratio, you will need to use your calculator. Most scientific and graphing calculators have buttons that look like \begin{align*}[\sin^{-1}], [\cos^{-1}]\end{align*}, and \begin{align*}[\tan^{-1}]\end{align*}. You will want to make sure that your calculator is in degree mode so that the angle measure that the calculator produces is in degrees.

Let's take a look at some example problems.

1. Solve for \begin{align*}\theta\end{align*}.

The side opposite the given angle is length 10 and the hypotenuse is length 22. This is a sine relationship.

\begin{align*}\sin \theta &= \frac{10}{22}\\ \theta &= \sin^{-1} \left(\frac{10}{22}\right)\\ \theta & \approx 27.04^\circ\end{align*}

2. Find \begin{align*}m \angle B\end{align*}.

The side opposite \begin{align*}\angle B\end{align*} is length 15 and the side adjacent to \begin{align*}\angle B\end{align*} is length 8. This is a tangent relationship.

\begin{align*}\tan B &= \frac{15}{8}\\ m \angle B &= \tan^{-1} \left(\frac{15}{8}\right)\\ m \angle B & \approx 61.93^\circ\end{align*}

3. Find \begin{align*}m \angle A\end{align*}.

You only need to know two sides of the triangle in order to find the measure of one of the angles. Since you are given all three sides, you can choose which two sides you want to use.

The side adjacent to \begin{align*}\angle A\end{align*} is length 12.7 and the hypotenuse is length 15. These two sides are a cosine relationship.

\begin{align*}\cos A &= \frac{12.7}{15}\\ m \angle A &= \cos^{-1} \left(\frac{12.7}{15}\right)\\ m \angle A & \approx 32.15^\circ\end{align*}

### Examples

#### Example 1

Earlier, you were asked what angle a ramp makes with the flat ground.

The maximum slope of a wheelchair ramp is 1:12. For a wheelchair ramp made with these specifications, what angle does the ramp make with the flat ground?

Draw a picture to represent this situation.

The side opposite the given angle is length 1 and the side adjacent to the given angle is length 12. This is a tangent relationship.

\begin{align*}\tan \theta &= \frac{1}{12}\\ \theta &= \tan^{-1} \frac{1}{12}\\ \theta &\approx 4.76^\circ\end{align*}

The wheelchair ramp makes approximately a \begin{align*}4.76^\circ\end{align*} angle with the ground.

#### Example 2

Find \begin{align*}m \angle A\end{align*}.

The given sides are opposite and adjacent to \begin{align*}\angle A\end{align*}. This is a tangent relationship.

\begin{align*}\tan A &= \frac{3.5}{6.2}\\ \angle A &= \tan^{-1} \frac{3.5}{6.2}\\ \angle A & \approx 29.45^\circ\end{align*}

#### Example 3

Find \begin{align*}m \angle B\end{align*}.

\begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are complementary:

\begin{align*}m \angle B &= 90^\circ-m \angle A\\ &= 90^\circ-29.45^\circ\\ &= 60.55^\circ\end{align*}

You could also use inverse tangent to find \begin{align*}m \angle B\end{align*}.

#### Example 4

Find \begin{align*}AB\end{align*}.

With all angle measures and two sides, you could use sine, cosine, or the Pythagorean Theorem to find \begin{align*}AB\end{align*}.  Using the Pythagorean Theorem:

\begin{align*}AB^2 &= 3.5^2+6.2^2\\ AB^2 &= 50.69\\ AB & \approx 7.12\end{align*}

### Review

1. What is the difference between \begin{align*}\sin^{-1}\end{align*} and \begin{align*}\sin\end{align*}?

2. When do you use regular trigonometric ratios and when do you use inverse trigonometric ratios?

Solve for \begin{align*}\theta\end{align*}.

3.

4.

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6.

7.

8.

9.

10. How could you have found \begin{align*}\theta\end{align*} in #9 without using an inverse trigonometric ratio?

Find all missing information (sides and angles) for each triangle.

11.

12.

13.

14. How are inverse cosine and inverse sine of the same ratio related?

15. How are inverse tangents of \begin{align*}a\end{align*} and \begin{align*}\frac{1}{a}\end{align*} related?

To see the Review answers, open this PDF file and look for section 7.4.

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### Vocabulary Language: English

TermDefinition
Adjacent Angles Two angles are adjacent if they share a side and vertex. The word 'adjacent' means 'beside' or 'next-to'.
Conic Conic sections are those curves that can be created by the intersection of a double cone and a plane. They include circles, ellipses, parabolas, and hyperbolas.
cosine The cosine of an angle in a right triangle is a value found by dividing the length of the side adjacent the given angle by the length of the hypotenuse.
Hypotenuse The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Pythagorean Theorem The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by $a^2 + b^2 = c^2$, where $a$ and $b$ are legs of the triangle and $c$ is the hypotenuse of the triangle.
sine The sine of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the hypotenuse.
Slope Slope is a measure of the steepness of a line. A line can have positive, negative, zero (horizontal), or undefined (vertical) slope. The slope of a line can be found by calculating “rise over run” or “the change in the $y$ over the change in the $x$.” The symbol for slope is $m$
Tangent The tangent of an angle in a right triangle is a value found by dividing the length of the side opposite the given angle by the length of the side adjacent to the given angle.