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# Applications of Sine

## Use a calculator to find sine of given angle

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Practice Applications of Sine
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Determine and Use the Sine Ratio

Mr. Watson’s homeroom decided to do a community service repairing the ramp outside the shed. The fresh coat of paint shone in the bright sunlight and Mr. Watson walked across the grass with all of his students to look at the ramp outside the door of the shed.

“Has that always been there?” asked Dan.

“No, in fact it was just brought out here yesterday,” Mr. Watson explained.

“Well if it’s brand new, then why do we need to fix it?” Emily asked.

“Because it doesn’t fit neatly under the door.”

Sure enough, the students looked and could see that the back of the ramp was too tall and the students would need to fix the back to make it fit beneath the doorway. Fixing this ramp would help everyone because it would make it easy to push or pull the cart with the athletic equipment out onto the field and since the equipment was used by local kid’s teams as well as the school, this was a definite way to give back to the community.

“What do we need to do?” Dan asked.

They looked at the ramp. Mr. Watson drew the following sketch.

“Well that’s not very helpful,” Dan commented.

“Sure it is,” Emily said.

This Concept is about using the sine ratio. By the end of the Concept, you will know how to figure out the height of the ramp.

### Guidance

A trigonometric ratio for a specific angle will remain constant no matter how large or small the triangle is. The idea is that the sides will always be in proportion to each other. So, if you know the measure of an angle (and can therefore identify the value of a trigonometric ratio) and the value of one side, you can use trigonometry to calculate the lengths of other sides.

The trick is to use good algebra technique, and make sure that every time you set up a ratio, you are putting the values and variables in the correct places.

You can find trigonometric ratios by using your calculator.

You understand trigonometric ratios and have had a chance to practice reading specific values out of a table.

You can find the ratio for any trigonometric value using your calculator. Take a moment to locate the buttons for sine, cosine, and tangent on the calculator. Keep in mind that usually, sine is abbreviated as sin, cosine is usually abbreviated as cos, and tangent is usually abbreviated as tan.

Press the key of the ratio you want to find, and enter the angle in question. If you hit enter, or calculate, the calculator will show you the value of that specific ratio.

Let's look at finding the sine ratio by using a calculator.

sine 47\begin{align*}\text{sine} \ 47^{\circ}\end{align*}

You can find the values for each ratio using your calculator. When dealing with large decimals values, it is usually best to round the numbers to the nearest thousandth. It gives you a reasonably accurate value without being too long of a number to work with.

The sine of 47\begin{align*}47^{\circ}\end{align*} is 0.73135370161917..., or about 0.731.

Notice that because we are using a calculator, that we don’t need to know the side lengths. The calculator figures out the trigonometric ratio based on proportions and the given angle. Figuring out sine, cosine and tangents using a calculator is as easy as pressing a button!

Now let's use the sine ratio when problem solving.

The ratio of sine is oppositehypotenuse\begin{align*}\frac{opposite}{hypotenuse}\end{align*}. If you know the sine value of the angle in question, and the length of the hypotenuse, you can find the measure of the opposite side. Look at the algebraic manipulation below.

sineXsineX×hypotenusesineX×hypotenuse=oppositehypotenuse=oppositehypotenuse×hypotenuse=opposite

If you multiply the sine of any angle X\begin{align*}X\end{align*} and the length of the hypotenuse, the result is the length of the opposite side.

Write this statement down in your notebook. Be sure that you write that it is connected to the sine ratio.

Now take a look at this situation.

What is the length of side AC\begin{align*}AC\end{align*} in the triangle below?

Use the following equation to find the length of the side opposite angle B\begin{align*}B\end{align*}.

Notice that we are going to multiply the sine of Angle B\begin{align*}B\end{align*} by the length of the hypotenuse to find the other side.

Therefore, you will have to find the sine of angle B\begin{align*}B\end{align*} first.

Then you multiply that times the length of the hypotenuse.

Notice that we are finding the length of the opposite side with sine.

sineB×hypotenusesine14.5×50.250×51.25=opposite=opposite=opposite=opposite

The length of side AC\begin{align*}AC\end{align*} is 1.25 units.

Use a calculator to determine each sine. You may round to the nearest hundredth.

#### Example A

Sine 45\begin{align*}\text{Sine} \ 45^{\circ}\end{align*}

Solution:  0.71\begin{align*}0.71\end{align*}

#### Example B

Sine 83\begin{align*}\text{Sine} \ 83^{\circ}\end{align*}

Solution:  0.99\begin{align*}0.99\end{align*}

#### Example C

Sine 27\begin{align*}\text{Sine} \ 27^{\circ}\end{align*}

Solution:  0.45\begin{align*}0.45\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

Now apply the sine ratio and figure out the height of the ramp.

First, we take the measurements and use the sine ratio.

Sine 15=OppositeHypotenuse\begin{align*}\text{Sine} \ 15^{\circ} = \frac{Opposite}{Hypotenuse}\end{align*}

The opposite in this example is the missing side. We use x\begin{align*}x\end{align*} to represent this unknown measure. This is the measure that we are searching for.

Sine 15=oppositehypotenuse=x4 ft\begin{align*}\text{Sine} \ 15^{\circ} = \frac{opposite}{hypotenuse}=\frac{x}{4 \ ft}\end{align*}

Now we can multiply both sides by 4 to find the measure.

4 Sine 152.6=x=x

The height of the ramp is 2.6 feet

### Vocabulary

Sine
a ratio between the opposite side and the hypotenuse of a given angle.
Cosine
a ratio between the adjacent side and the hypotenuse of a given angle.
Tangent
a ratio between the opposite side and the adjacent side of a given angle.
Trigonometric Ratio
used to find missing side lengths of right triangles when angle measures have been given.

### Guided Practice

Here is one for you to try on your own.

Find the length of the hypotenuse.

Solution

First, here is the ratio that we are going to be using.

Sine 45=OppositeHypotenuse\begin{align*}\text{Sine} \ 45^{\circ} = \frac{Opposite}{Hypotenuse}\end{align*}

The hypotenuse in this example is the missing side. We use x\begin{align*}x\end{align*} to represent this unknown measure. This is the measure that we are searching for.

Sine 45=oppositehypotenuse=4x\begin{align*}\text{Sine} \ 45^{\circ} = \frac{opposite}{hypotenuse}=\frac{4}{x}\end{align*}

Now we can write the following equation.

xSine 45=4\begin{align*}x\text{Sine} \ 45^{\circ} = 4\end{align*}

Next, find the Sine of 45 = .8 = .71

.71xx=4=5.6

### Practice

Directions: Use a calculator to find each Sine. You may round to the nearest hundredth.

1. Sine 55\begin{align*}\text{Sine} \ 55^{\circ}\end{align*}
2. Sine 25\begin{align*}\text{Sine} \ 25^{\circ}\end{align*}
3. Sine 11\begin{align*}\text{Sine} \ 11^{\circ}\end{align*}
4. Sine 60\begin{align*}\text{Sine} \ 60^{\circ}\end{align*}
5. Sine 75\begin{align*}\text{Sine} \ 75^{\circ}\end{align*}
6. \begin{align*}\text{Sine} \ 12^{\circ}\end{align*}
7. \begin{align*}\text{Sine} \ 29^{\circ}\end{align*}
8. \begin{align*}\text{Sine} \ 15^{\circ}\end{align*}

Directions: Use the information given and what you have learned about trigonometric ratios to figure out the measure of each missing side. You may round when necessary.

1. Sine angle \begin{align*}D \ 2^{\circ}\end{align*}, hypotenuse – 12, what is the length of the opposite side?
2. Sine angle \begin{align*}E \ 65^{\circ}\end{align*}, hypotenuse – 8, what is the length of the opposite side?
3. Sine angle \begin{align*}F \ 45^{\circ}\end{align*}, hypotenuse – 2, what is the length of the opposite side?
4. Sine angle \begin{align*}D \ 25^{\circ}\end{align*}, hypotenuse – 10, what is the length of the opposite side?
5. Sine angle \begin{align*}D \ 80^{\circ}\end{align*}, hypotenuse – 8, what is the length of the opposite side?
6. Sine angle \begin{align*}D \ 45^{\circ}\end{align*}, hypotenuse – 5, what is the length of the opposite side?
7. Sine angle \begin{align*}D \ 40^{\circ}\end{align*}, hypotenuse – 18, what is the length of the opposite side?

### Vocabulary Language: English

cosine

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.
sine

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.
Tangent

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.
Trigonometric Ratios

Trigonometric Ratios

Ratios that help us to understand the relationships between sides and angles of right triangles.