Lesson 8 Are Relationships Predictable? Develop Understanding

Jump Start

In the previous lesson, one possible arrangement of the right triangles may have looked like this:

1.

How does this arrangement of the triangles suggest that the three triangles are similar?

2.

How did we justify that the triangles were similar?

3.

If we had chosen to use a different-sized rectangle, such as a $3 \times 5$ card, or a $5 \times 8$ card, or an $8.5 \times 11 inch$ sheet of paper, and cut along the diagonal to form the hypotenuse of the largest triangle, and then cut the second triangle formed by that same rectangle along the altitude drawn from the right angle to form two smaller triangles, would we have been able to arrange the three triangles in this same way? Why or why not?

Learning Focus

Investigate corresponding ratios of right triangles with the same acute angle.

What determines if two right triangles are similar?

Why are the ratios of sides in right triangles so special that they deserve a classification of their own (trigonometric ratios)?

Open Up the Math: Launch, Explore, Discuss

Draw a right triangle with an acute angle that measures $60 °$ .

Measure each side of your triangle as accurately as you can with a centimeter ruler.

Using the $60 °$ angle as the angle of reference, list the measure for each of the following:

Length of the adjacent side:

Length of the opposite side:

Length of the hypotenuse:

Create the following ratios using your measurements:

$\frac{opposite side}{hypotenuse} =$

$\frac{adjacent side}{hypotenuse} =$

$\frac{opposite side}{adjacent side} =$

1.

Compare your ratios with others that had a triangle of a different size. What do you notice? Explain any connections you find to others’ work.

2.

In the right triangles provided, find the missing side length and then create the desired ratios based on the angle of reference (angle $A$ and angle $D$ ).

List the ratios for $△ A B C$ using angle $A$ as the angle of reference.

$\frac{opposite side}{hypotenuse} = \underset{―}{}$

$\frac{adjacent side}{hypotenuse} = \underset{―}{}$

$\frac{opposite side}{adjacent side} = \underset{―}{}$

List the ratios for $△ D E F$ using angle $D$ as the angle of reference.

$\frac{opposite side}{hypotenuse} = \underset{―}{}$

$\frac{adjacent side}{hypotenuse} = \underset{―}{}$

$\frac{opposite side}{adjacent side} = \underset{―}{}$

3.

What do you notice about the ratios from the two given triangles? How do these ratios compare to the ratios from the triangle you made for problem 1?

4.

What can you infer about the angle measures of $△ A B C$ and $△ D E F$ ?

5.

Why do the relationships you have noticed occur?

6.

Measure the angles in both $△ A B C$ and $△ D E F$ . What can you conclude about the ratio of sides of a right triangle that have these specific angles?

7.

What might you conclude about the ratio of sides of a right triangle that has a $45 °$ angle? If necessary, draw and experiment with examples of such a triangle.

The ratios we have worked with in this task are given special names.

The sine is the ratio of the length of the side opposite an angle to the length of the hypotenuse.
The cosine is the ratio of the length of the side adjacent to an angle to the length of the hypotenuse.
The tangent is the ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.

Ready for More?

Use the diagram to make observations about the sine, cosine and tangent ratios to answer the following questions:

a.

As the reference angle $A$ gets bigger, will the value of the sine ratio increase or decrease?

b.

What happens to the value of the sine ratio when the reference angle $A$ is close to $0 °$ ?

c.

What happens to the value of the sine ratio when the reference angle $A$ is close to $90 °$ ?

d.

As the reference angle $A$ gets bigger, will the value of the cosine ratio increase or decrease?

e.

What happens to the value of the cosine ratio when the reference angle $A$ is close to $0 °$ ?

f.

What happens to the value of the cosine ratio when the reference angle $A$ is close to $90 °$ ?

g.

As the reference angle $A$ gets bigger, will the value of the tangent ratio increase or decrease?

h.

What happens to the value of the tangent ratio when the reference angle $A$ is close to $0 °$ ?

i.

What happens to the value of the tangent ratio when the reference angle $A$ is close to $45 °$ ?

j.

What happens to the value of the tangent ratio when the reference angle $A$ is close to $90 °$ ?

Takeaways

The ratios of sides of a right triangle are called trigonometric ratios, and each specific ratio is given a name.

The ratio $\frac{opposite side}{hypotenuse}$ is called .

The ratio $\frac{adjacent side}{hypotenuse}$ is called .

The ratio $\frac{opposite side}{adjacent side}$ is called .

The measure of the acute reference angle in a right triangle determines the value of these ratios, because

Before calculators, these values were recorded in tables for reference. For each, a table of trigonometric ratios would include these values:

$\sin (60 °) =$ $\underset{―}{}$ $\cos (60 °) =$ $\underset{―}{}$ $\tan (60 °) =$ $\underset{―}{}$

$\sin (30 °) =$ $\underset{―}{}$ $\cos (30 °) =$ $\underset{―}{}$ $\tan (30 °) =$ $\underset{―}{}$

$\sin (45 °) =$ $\underset{―}{}$ $\cos (45 °) =$ $\underset{―}{}$ $\tan (45 °) =$ $\underset{―}{}$

Adding Notation, Vocabulary, and Conventions

In a right triangle, we have given the name to the side opposite the right angle. It is always the longest side of the right triangle.

We will also give special names to the two legs of a right triangle, relative to one of the acute angles of the triangle.

In the diagram, $\overset{―}{A C}$ is the side to $∠ A$ and $\overset{―}{B C}$ is the side $∠ A$ .

In the diagram, $\overset{―}{A C}$ is the side $∠ B$ and $\overset{―}{B C}$ is the side to $∠ B$ .

I can quickly recognize how to name each of the following sides of a right triangle by…

Hypotenuse:

Adjacent side:

Opposite side:

Vocabulary

adjacent
hypotenuse
opposite side in a triangle
reference angle
trigonometric ratios in right triangles: sine A, cosine A, tangent A
Bold terms are new in this lesson.

Lesson Summary

In this lesson, we learned about some special ratios, called trigonometric ratios, that occur in right triangles. If two right triangles have a pair of corresponding acute angles that are congruent, the right triangles will be similar. Therefore, corresponding ratios of the sides of these two right triangles will be equal. This observation is so useful when working with right triangles that have the same acute angle that values of these ratios were recorded in tables for each acute angle between $0 °$ and $90 °$ .

Retrieval

Find the missing angle measurement and side length in each right triangle.

1.

2.

Find the factored form and the intercepts for each quadratic function.

3.

$g (x) = x^{2} - 4 x - 21$

Factored form:

$y$ -intercept:

$x$ -intercepts:

4.

$h (x) = x^{2} + 8 x + 12$

Factored form:

$y$ -intercept:

$x$ -intercepts: