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# 7.7: Exploring Trigonometric Ratios

Difficulty Level: At Grade Created by: CK-12

## Introduction

The Ramp Dilemma

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.

Who is right? Is it possible to figure out the length of the missing side using Mr. Watson’s diagram? What kind of math will you need?

What You Will Learn

In this lesson, you will learn how to complete and understand the following tasks.

• Recognize the sine of a given acute angle of a right triangle as the ratio of the length of the opposite leg to the length of the hypotenuse.
• Recognize the cosine of a given acute angle of a right triangle as the ratio of the length of the adjacent leg to the length of the hypotenuse.
• Recognize the tangent ratio of a given acute angle of a right triangle as the ratio of the length of the opposite leg to the length of the adjacent leg.
• Find sine, cosine, and tangent ratios for each of the acute angles of right triangles with given side lengths.
• Use tables to identify and understand patterns of sine, cosine and tangent ratios as angles increase from 0 to \begin{align*}90^\circ\end{align*}.

Teaching Time

I. Understanding and Recognize the Sine of a Right Triangle

One way to analyze right triangles is through trigonometric ratios. There are three trigonometric ratios and they help us to understand the proportions between the sides and the angles. Pay special attention to the specifics of each ratio, as you will have to remember these.

Let’s look at the first trigonometric ratio. It is called the sine. A sine refers to a particular angle in a right triangle. The sine of an angle is the ratio of the length of the leg opposite the angle we are focusing on to the length of the hypotenuse. Remember that in a ratio, you list the first item on top of the fraction and the second item on the bottom. So, the ratio of the sine will be \begin{align*}\frac{opposite}{hypotenuse}\end{align*}.

Let’s look at how we can find the sine of a particular angle.

Example

What are the sines of \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} in the triangle below?

To find the sine, all you have to do to is build the ratio carefully.

\begin{align*}\sin e \angle A & = \frac{opposite}{hypotenuse} = \frac{3}{5} = 0.6\\ \sin e \angle B & = \frac{opposite}{hypotenuse} = \frac{4}{5} = 0.8\end{align*}

Notice that once we have the ratio, that we can divide the numerator by the denominator to convert it to a decimal. The decimal is the answer that we are looking for with regard to the trigonometric ratio.

The sine of \begin{align*}\angle A\end{align*} is 0.6 and the sine of \begin{align*}\angle B\end{align*} is 0.8.

As you could see in example, the side opposite an angle is the one that an angle opens up to. An opposite side will never be one of the rays that forms and angle.

II. Recognize the Cosine of a Given Acute Angle of a Right Triangle as the Ratio of the Length of the Adjacent Leg to the Length of the Hypotenuse

The next ratio to examine is called the cosine. The cosine is the ratio of the adjacent side of an angle to the hypotenuse. Use the same techniques you used to find sines to find cosines.

Example

What are the cosines of \begin{align*}\angle M\end{align*} and \begin{align*}\angle N\end{align*} in the triangle below

To find these ratios, identify the sides adjacent to each angle and the hypotenuse. Remember that an adjacent side is the one that does create the angle and is not the hypotenuse.

\begin{align*}\cos ine \angle M & = \frac{adjacent}{hypotenuse} = \frac{15}{17} \approx 0.88\\ \cos ine \angle N & = \frac{adjacent}{hypotenuse} = \frac{8}{17} \approx 0.47\end{align*}

Once again notice that we divided the numerator by the denominator to find a decimal representation of the cosine of each of the angles. You can figure these ratios out on your calculator.

The cosine of \begin{align*}\angle M\end{align*} is about 0.88 and the cosine of \begin{align*}\angle N\end{align*} is about 0.47.

III. Recognize the Tangent Ratio of a Given Acute Angle of a Right Triangle as the Ratio of the Length of the Opposite Leg to the Length of the Adjacent Leg

The final ratio to examine when studying right triangles is the tangent. The tangent is the ratio of the opposite side to the adjacent side of an angle. The hypotenuse is not involved in the tangent at all.

Example

What are the tangents of \begin{align*}\angle X\end{align*} and \begin{align*}\angle Y\end{align*} in the triangle below?

To find these ratios, first identify the sides opposite and adjacent to each angle.

\begin{align*}\tan gent \angle X & = \frac{opposite}{adjacent} = \frac{5}{12} \approx 0.417\\ \tan gent \angle Y & = \frac{opposite}{adjacent} = \frac{12}{5} = 2.4\end{align*}

The tangent of \begin{align*}\angle X\end{align*} is about 0.417 and the tangent of \begin{align*}\angle Y\end{align*} is 2.4.

IV. Find Sine, Cosine and Tangent Ratios for Each of the Acute Angles of Right Triangles with Given Side Lengths

Now that you understand how to find the sine, cosine and tangent of a right triangle, we can use this information to find the sine, cosine and tangent of each of the acute angles of right triangles.

Let’s look at an example.

Example

Find the sine, cosine and tangent of each acute angle in the triangle below.

First, we need to identify the acute angles. We know that an acute angle is less than \begin{align*}90^\circ\end{align*}, so in this example, \begin{align*}\angle A\end{align*} and \begin{align*}\angle C\end{align*} are the acute angles. We will find the sine, cosine and tangent of each of these angles.

Let’s start with the sine ratio for each. We know that the sine is the opposite over the hypotenuse. Here are the sine ratios.

\begin{align*}\text{Sine} \angle A & = \frac{8}{10} = .8\\ \text{Sine} \angle C & = \frac{6}{10} = .6\end{align*}

Now let’s look at the cosine of each. Cosine is the ratio of the adjacent side to the hypotenuse. Here are the cosine ratios.

\begin{align*}\text{Cosine} \angle A & = \frac{6}{10} = .6\\ \text{Cosine} \angle C &= \frac{8}{10} = .8\end{align*}

Next, we can find the tangent ratios. Tangent is the ratio of the opposite side to the adjacent side. Here are the tangent ratios.

\begin{align*}\text{Tangent} \angle A & = \frac{8}{6} = 1.333\\ \text{Tangent} \angle C & = \frac{6}{8} = .75\end{align*}

Notice that figuring out the ratios is not that difficult. The key is to be sure that you have memorized the different ratios.

Write each of these ratios down in your notebook.

V. Use Tables to Identify and Understand Patterns of Sine, Cosine and Tangent Ratios as Angles Increase from 0 to \begin{align*}\underline{90^\circ}\end{align*}

You’ll learn in the next lesson how to best use table and calculators to understand trigonometric ratios. Here you can get a preview about how the values change. This lesson focused primarily on how to identify the different ratios, but each ratio will have a constant value for a specific angle. In any right triangle, the sine of the \begin{align*}30^\circ\end{align*} angle will always be 0.5. You can use that information to find missing lengths in triangle where you know the angles, or to identify the measure of an angle if you know two of the sides.

Examine the table below for trends and patterns. This table shows the sine, cosine, and tangent values for eight different angles measures.

\begin{align*}10^\circ\end{align*} \begin{align*}20^\circ\end{align*} \begin{align*}30^\circ\end{align*} \begin{align*}40^\circ\end{align*} \begin{align*}50^\circ\end{align*} \begin{align*}60^\circ\end{align*} \begin{align*}70^\circ\end{align*} \begin{align*}80^\circ\end{align*}
sine 0.174 0.342 0.5 0.643 0.766 0.866 0.940 0.985
cosine 0.985 0.940 0.866 0.766 0.643 0.5 0.342 0.174
tangent 0.176 0.364 0.577 0.839 1.192 1.732 2.747 5.671

Example

Using the table above, which value would you expect to be greater: the sine of \begin{align*}25^\circ\end{align*} or the cosine of \begin{align*}25^\circ\end{align*}?

You can use the information in the table to solve this problem. The sine of \begin{align*}20^\circ\end{align*} is 0.342 and the sine of \begin{align*}30^\circ\end{align*} is 0.5. So, the sine of \begin{align*}25^\circ\end{align*} will be between the values 0.342 and 0.5. The cosine of \begin{align*}20^\circ\end{align*} is 0.940 and the cosine of \begin{align*}30^\circ\end{align*} is 0.866. So, the cosine of \begin{align*}25^\circ\end{align*} will be between the values of 0.866 and 0.940. Since the range for the cosine is greater, than the range for the sine, it can be assumed that the cosine of \begin{align*}25^\circ\end{align*} will be greater than the sine of \begin{align*}25^\circ\end{align*}.

Notice that as the angle measures approach \begin{align*}90^\circ\end{align*}, sine approaches the value of 1. Similarly, as the value of the angles approaches \begin{align*}90^\circ\end{align*}, the cosine approaches the value of 0. In other words, as the sine gets greater, the cosine loses value.

The tangent, on the other hand, increases rapidly from a small value to a large value approaching \begin{align*}90^\circ\end{align*}.

## Real-Life Example Completed

The Ramp Dilemma

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.

Who is right? Is it possible to figure out the length of the missing side using Mr. Watson’s diagram? What kind of math will you need?

Solution to Real – Life Example

In thinking about this problem, Emily is correct. You can use the diagram to solve the problem. To do this, you will need to use trigonometric ratios. In fact, you will need to use the sine ratio to figure out the height of the ramp. That is the work of the next lesson.

## Vocabulary

Here are the vocabulary words that are found in this lesson.

Trigonometric Ratios
ratios that help us to understand the relationships between sides and angles of right triangles.
Sine
the ratio of the opposite side to the hypotenuse.
Cosine
the ratio of the adjacent side to the hypotenuse.
Tangent
the ratio of the opposite side to the adjacent.

## Time to Practice

Directions: solve each problem.

Use the following diagram for problems 1-9

1. What is the sin of \begin{align*}\angle G\end{align*}?
2. What is the cosine of \begin{align*}\angle G\end{align*}?
3. What is the tangent of \begin{align*}\angle H\end{align*}?
4. What is the sin of \begin{align*}\angle H\end{align*}?
5. What is the cosine of \begin{align*}\angle H\end{align*}?
6. What is the tangent of \begin{align*}\angle G\end{align*}?
7. What is the sine of \begin{align*}\angle A\end{align*}?
8. What is the cosine of \begin{align*}\angle A\end{align*}?
9. What is the tangent of \begin{align*}\angle A\end{align*}?

Use the following diagram for problems 10-13.

1. What is the tangent of \begin{align*}\angle R\end{align*}?
2. What is the tangent of \begin{align*}\angle S\end{align*}?
3. What is the sine of \begin{align*}\angle R\end{align*}?
4. What is the cosine of \begin{align*}\angle S\end{align*}?

Use the following table for problems 14 - 21.

\begin{align*}10^\circ\end{align*} \begin{align*}20^\circ\end{align*} \begin{align*}30^\circ\end{align*} \begin{align*}40^\circ\end{align*} \begin{align*}50^\circ\end{align*} \begin{align*}60^\circ\end{align*} \begin{align*}70^\circ\end{align*} \begin{align*}80^\circ\end{align*}
sine 0.174 0.342 0.5 0.643 0.766 0.866 0.940 0.985
cosine 0.985 0.940 0.866 0.766 0.643 0.5 0.342 0.174
tangent 0.176 0.364 0.577 0.839 1.192 1.732 2.747 5.671
1. Which value will be greater, the tangent of \begin{align*}45^\circ\end{align*} or the cosine of \begin{align*}45^\circ\end{align*}?
2. Which value will be greater, the tangent of \begin{align*}55^\circ\end{align*} or the sine of \begin{align*}55^\circ\end{align*}?
3. Which value will be greater, the sine of \begin{align*}85^\circ\end{align*} or the cosine of \begin{align*}85^\circ\end{align*}?
4. What is the tangent of a \begin{align*}60^\circ\end{align*} angle?
5. What is the sine of a \begin{align*}60^\circ\end{align*} angle?
6. What is the cosine of a \begin{align*}10^\circ\end{align*} angle?
7. What is the cosine of a \begin{align*}30^\circ\end{align*} angle?
8. What is the sine of an \begin{align*}80^\circ\end{align*} angle?

1. What is the ratio for sine?
2. What is the ratio for cosine?
3. What is the ratio for tangent?

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