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?

Circle, A. Triangle with vertex A, vertex B on the circle, vertex C inside the circle. Angle C is a right angle. Corresponding pposite sides of the triangle are labeled with lower-case a, b c.

 

2.2: Recalling Right Triangle Trigonometry

  1. Find \(\cos(A)\), \(\sin(A)\), and \(\tan(A)\) for triangle \(ABC\).
    Triangle A, B C. Angle B is a right angle. Side A, B length 15, side C B length 8, side A, C length 17.
  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.
    A circle on centered on a coordinate plane. Origin is labeled G = 0 comma 0. Point H at 9 comma 0. Point I lies  vertically above H and is on the circle. Triangle G H I is drawn with H a right angle.

2.3: Shrinking Triangles

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

    Coordinate plane, x 0 to 12 by 2, y, 0 to 6 by 2. Triangle drawn from D at 0 comma 0 to E at 12 comma 0 to F at 12 comma 5. Angle E is a right angle.

  2. Here is a triangle similar to triangle \(DEF\).

    Coordinate plane, x 0 to 36 by 3, y, 0 to 18 by 3. Triangle drawn from D prime at 0 comma 0 to E prime at 36 comma 0 to F prime at 36 comma 15. Angle E prime is a right angle.
    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 \( DEF\).

    Coordinate plane, x, 0 to 1 by point 5, y, 0 to 1 by point 5. A triangle, one vertex on the origin, one vertex on x axis between point 5 & 1, and the third vertex directly above. Hypotenuse length 1.
    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.

Triangle A, B C. C is a right angle. Side A, C length 4, side B C length 3, side A, B length 5.

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:

Coordinate plane, x, 0 to 1 by point 5, y, 0 to 1 by point 5. Triangle, A, prime at the origin, C prime on the x axis between point 5 and 1, B prime above C prime with y value between point 5 and 1.

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 \(x\).