Lesson 2

Revisiting Right Triangles

• Let’s recall and use some things we know about right triangles.

2.1: Notice and Wonder: A Right Triangle

What do you notice? What do you wonder?

2.2: Recalling Right Triangle Trigonometry

1. Find $$\cos(A)$$, $$\sin(A)$$, and $$\tan(A)$$ for triangle $$ABC$$.
2. Sketch a triangle $$DEF$$ where $$\sin(D)=\cos(D)$$ and $$E$$ is a right angle. What is the value of $$\tan(D)$$ for this triangle? Explain how you know.
3. If the coordinates of point $$I$$ are $$(9,12)$$, what is the value of $$\cos(G)$$, $$\sin(G)$$, and $$\tan(G)$$ for triangle $$GHI$$? Explain or show your reasoning.

2.3: Shrinking Triangles

1. What are $$\cos(D)$$, $$\sin(D)$$, and $$\tan(D)$$? Explain how you know.

2. Here is a triangle similar to triangle $$DEF$$.

1. What is the scale factor from $$\triangle DEF$$ to $$\triangle D'E'F'$$? Explain how you know.
2. What are $$\cos(D')$$, $$\sin(D')$$, and $$\tan(D')$$?
3. Here is another triangle similar to triangle $$DEF$$.

1. Label the triangle $$D’'E’'F’'$$.
2. What is the scale factor from triangle $$DEF$$ to triangle $$D’'E’'F’'$$?
3. What are the coordinates of $$F’'$$? Explain how you know.
4. What are $$\cos(D'')$$, $$\sin(D'')$$, and $$\tan(D'')$$?

Angles $$C$$ and $$C’$$ in triangles $$ABC$$ and $$A’B’C’$$ are right angles. If $$\sin(A) = \sin(A’)$$, is that sufficient to show that $$\triangle ABC$$ is similar to $$\triangle A’B’C’$$? Explain your reasoning.

Summary

In an earlier course, we studied ratios of side lengths in right triangles.

In this triangle, the cosine of angle $$A$$ is the ratio of the length of the side adjacent to angle $$A$$ to the length of the hypotenuse—that is $$\cos(A) = \frac{4}{5}$$. The sine of angle $$A$$ is the ratio of the length of the side opposite angle $$A$$ to the length of the hypotenuse—that is $$\sin(A) = \frac{3}{5}$$. The tangent of angle $$A$$ is the ratio of the length of the side opposite angle $$A$$ to the length of the side adjacent to angle $$A$$—that is $$\tan(A) = \frac{3}{4}$$.

Now consider triangle $$A’B’C’$$, which is similar to triangle $$ABC$$ with a hypotenuse of length 1 unit. Here is a picture of triangle $$A’B’C’$$ on a coordinate grid:

Since the two triangles are similar, angle $$A$$ and $$A'$$ are congruent. So how do the values of cosine, sine, and tangent of these angles compare to the angles in triangle $$ABC$$? It turns out that since all three values are ratios of side lengths, $$\cos(A)=\cos(A')$$, $$\sin(A)=\sin(A')$$, and $$\tan(A)=\tan(A')$$.

Notice that the coordinates of $$B’$$ are $$\left(\frac{4}{5},\frac{3}{5}\right)$$ because segment $$A’C’$$ has length $$\frac{4}{5}$$ and segment $$B’C’$$ has length $$\frac{3}{5}$$. In other words, the coordinates of $$B’$$ are $$(\cos(A'),\sin(A'))$$.

Glossary Entries

• period

The length of an interval at which a periodic function repeats. A function $$f$$ has a period, $$p$$, if $$f(x+p) = f(x)$$ for all inputs $$x$$.

• periodic function

A function whose values repeat at regular intervals. If $$f$$ is a periodic function then there is a number $$p$$, called the period, so that $$f(x + p) = f(x)$$ for all inputs $$x$$.