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Precalculus for Team-Based Inquiry Learning 2025 Edition

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Section 6.4 Special Right Triangles (TR4)

Subsection 6.4.1 Activities

Remark 6.4.1.

Recall from the previous section that we can find values of trigonometric functions of any angle. In this section, we can take what we know about right triangles and find exact values of trigonometric functions of special angles.
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Definition 6.4.2.

There are two special right triangle relationships that will continually appear when speaking about special angles.
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Activity 6.4.3.

Let’s explore the relationship of the \(45-45-90\) special right triangle.
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(a)
Draw a right triangle and label the angles to have \(45\)°, \(45\)°, and \(90\)°.
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(b)
In a \(45-45-90\) triangle, two angles are the same size. If those two angles are the same size, what do we know about the sides opposite those angles?
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(c)
Suppose one of the legs of the right triangle is of length \(1\text{,}\) how long is the other leg?
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(d)
Now that we know two sides of the right triangle, use that information and the Pythagorean Theorem to find the length of the third side.
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Definition 6.4.4.

From Activity 6.4.3, we saw that a \(45-45-90\) triangle is an isosceles right triangle, which means that two of the sides of the triangle are equal. The ratio of its legs and hypotenuse is expressed as follows:
\begin{equation*} \text{Leg}:\text{Leg}:\text{Hypotenuse}=1:1:\sqrt{2}\text{.} \end{equation*}
In terms of \(x\text{,}\) this ratio can be expressed as
\begin{equation*} x:x:x\sqrt{2}\text{.} \end{equation*}
Therefore, the 45-45-90 triangle rule states that the three sides of the triangle are in the ratio \(x:x:x\sqrt{2}\text{.}\)
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Activity 6.4.5.

Suppose you are given a right triangle \(ABC\text{,}\) where \(C=45\)°, \(a=6\) cm, and \(b\) is the hypotenuse.
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(a)
Draw a picture of this right triangle and label the sides.
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Activity 6.4.6.

For each of the following, use Definition 6.4.4 to find the missing side.
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(a)
In triangle \(ABC\text{,}\) \(A=90^\circ\text{,}\) \(c=3\text{,}\) and \(B=45^\circ\text{.}\) What is the length of \(a\text{?}\)
  1. \(\displaystyle 6\)
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  2. \(\displaystyle 5\)
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  3. \(\displaystyle 3\)
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  4. \(\displaystyle 3\sqrt{2}\)
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(b)
In triangle \(ABC\text{,}\) \(B=90^\circ\text{,}\) \(b=6\sqrt{2}\text{,}\) and \(C=45^\circ\text{.}\) What is the length of \(c\text{?}\)
  1. \(\displaystyle 6\)
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  2. \(\displaystyle 5\)
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  3. \(\displaystyle 4\)
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  4. \(\displaystyle 12\)
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(c)
In triangle \(ABC\text{,}\) \(B=90^\circ\text{,}\) \(b=4\text{,}\) and \(A=45^\circ\text{.}\) What is the length of \(c\text{?}\)
  1. \(\displaystyle \sqrt{6}\)
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  2. \(\displaystyle 2\sqrt{2}\)
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  3. \(\displaystyle 4\)
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  4. \(\displaystyle 4\sqrt{2}\)
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Activity 6.4.7.

Suppose you are given an equilateral triangle, which has three equal sides and three equal angles (\(60\)°).
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(a)
Draw an equilateral triangle and then draw the height from the base of the triangle to the top angle. What kind of triangles did you just create when drawing the height?
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(b)
What do you notice about the relationships of the sides of the equilateral triangle to that of the \(30-60-90\) triangles?
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(c)
Label the angles to have \(30\)°, \(60\)°, and \(90\)° and the side opposite of the \(30\)° angle as having a length of \(1\text{.}\)
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(d)
Given that the side opposite the \(30\)° angle has a length of \(1\text{,}\) how long is the length of one side of the equilateral triangle?
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(e)
Now that you know the length of two sides of the \(30-60-90\) triangles, find the length of the third side using the Pythagorean Theorem.
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Definition 6.4.8.

From Activity 6.4.7, we saw that if a triangle has angle measures \(30\)°, \(60\)°, and \(90\)°, then the sides are in the ratio:
\begin{equation*} 1:\sqrt{3}:2\text{.} \end{equation*}
In terms of \(x\text{,}\) this ratio can be expressed as
\begin{equation*} x:x\sqrt{3}:2x\text{.} \end{equation*}
Therefore, the 30-60-90 triangle rule states that the three sides of the triangle are in the ratio \(x:x\sqrt{3}:2x\text{.}\) Note that the shorter leg is always \(x\text{,}\) the longer leg is always \(x\sqrt{3}\text{,}\) and the hypotenuse is always \(2x\text{.}\)
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Activity 6.4.9.

Suppose you are given a right triangle \(ABC\text{,}\) where \(C=30\)°, \(c=7\) cm, and \(b\) is the hypotenuse.
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(a)
Draw a picture of this right triangle and label the sides.
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Activity 6.4.10.

For each of the following, use Definition 6.4.8 to find the missing side.
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(a)
In triangle \(ABC\text{,}\) \(B=90\text{,}\)°, \(c=6\text{,}\) and \(C=30\)°. What is the length of \(a\text{?}\)
  1. \(\displaystyle 6\sqrt{3}\)
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  2. \(\displaystyle 4\)
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  3. \(\displaystyle 3\)
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  4. \(\displaystyle 6\sqrt{2}\)
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(b)
In triangle \(ABC\text{,}\) \(C=90\text{,}\)°, \(a=4\sqrt{3}\text{,}\) and \(B=30\)°. What is the length of \(b\text{?}\)
  1. \(\displaystyle 3\)
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  2. \(\displaystyle 6\)
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  3. \(\displaystyle 4\sqrt{2}\)
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  4. \(\displaystyle 4\)
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(c)
In triangle \(ABC\text{,}\) \(B=90\text{,}\)°, \(a=8\sqrt{3}\text{,}\) and \(A=60\)°. What is the length of \(b\text{?}\)
  1. \(\displaystyle 4\)
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  2. \(\displaystyle 16\)
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  3. \(\displaystyle 12\)
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  4. \(\displaystyle 8\sqrt{2}\)
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Remark 6.4.11.

Recall that we can find values of trigonometric functions of angles of right triangles. We can use the same idea to find values of trigonometric functions of angles of special right triangles.
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Activity 6.4.12.

Suppose you are given triangle \(ABC\text{,}\) where \(C=45\)°, \(a=5\text{,}\) and \(b\) is the hypotenuse.
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(a)
Find the measures of sides \(b\) and \(c\text{.}\)
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(b)
Now that you have found all the sides and angles of triangle \(ABC\text{,}\) find the ratio that represents \(\tan{A}\text{.}\)
  1. \(\displaystyle \frac{5}{5\sqrt{2}}\)
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  2. \(\displaystyle \frac{5\sqrt{2}}{5}\)
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  3. \(\displaystyle \sqrt{2}\)
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  4. \(\displaystyle 1\)
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(c)
Find the ratio that represents \(\sin{A}\text{.}\)
  1. \(\displaystyle \frac{5}{5\sqrt{2}}\)
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  2. \(\displaystyle \frac{5\sqrt{2}}{5}\)
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  3. \(\displaystyle \frac{1}{\sqrt{2}}\)
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  4. \(\displaystyle \frac{\sqrt{2}}{2}\)
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(d)
What is the approximate value of \(\sin{A}\text{?}\)
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Remark 6.4.13.

In Activity 6.4.12, notice that when finding a value of a trigonometric ratio, sometimes the values are decimal approximations. For example, the ratio for \(\sin{A}\) was \(\frac{1}{\sqrt{2}}\) (or \(\frac{\sqrt{2}}{2}\)), which gives an approximate value of \(0.707\text{.}\)
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Definition 6.4.14.

The exact values of trigonometric functions are values of trigonometric functions of certain angles that can be expressed exactly using expressions containing real numbers and roots of real numbers. When finding trigonometric ratios, we often give an exact value, rather than an approximation.
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Activity 6.4.15.

Find the exact value of the trigonometric function you are asked to find for each of the following.
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(a)
Given the triangle, \(ABC\text{,}\) with \(B=90\)°, \(C=45\)°, and \(a=7\text{,}\) find \(\cos{A}\text{.}\)
  1. \(\displaystyle \frac{7}{7\sqrt{2}}\)
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  2. \(\displaystyle \frac{7\sqrt{2}}{7}\)
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  3. \(\displaystyle \frac{1}{\sqrt{2}}\)
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  4. \(\displaystyle \frac{\sqrt{2}}{2}\)
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(b)
Using your calculator, find the value of \(\cos{45}\)° to the nearest thousandth.
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(c)
Which of the following is equivalent to the value of \(\cos{45}\)° that you found in part (b)?
  1. \(\displaystyle \frac{7}{7\sqrt{2}}\)
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  2. \(\displaystyle \frac{7\sqrt{2}}{7}\)
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  3. \(\displaystyle \frac{1}{\sqrt{2}}\)
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  4. \(\displaystyle \frac{\sqrt{2}}{2}\)
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(d)
How can using the \(45-45-90\) triangle help us find the value of \(\sin{45}\)°?
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Remark 6.4.16.

Notice that if you know the relationships of the sides of special right triangles, it can help you find the exact value of special angles (i.e., \(30\)°, \(45\)°, and \(60\)°). In addition, notice that the trig values of the angles were the same regardless of the size of the triangle.
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Activity 6.4.17.

For each of the following, find the exact value of the trigonometric function. Use the \(45-45-90\) and \(30-60-90\) trigonometric rules to help you:
\begin{equation*} 1:1:\sqrt{2} \end{equation*}
\begin{equation*} 1:\sqrt{3}:2 \end{equation*}
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(a)
What is \(\tan45\)°?
  1. \(\displaystyle \frac{1}{\sqrt{2}}\)
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  2. \(\displaystyle \sqrt{2}\)
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  3. \(\displaystyle 1\)
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  4. \(\displaystyle \frac{\sqrt{2}}{2}\)
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(b)
What is \(\cos60\)°?
  1. \(\displaystyle \frac{1}{2}\)
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  2. \(\displaystyle \sqrt{3}\)
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  3. \(\displaystyle 2\)
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  4. \(\displaystyle \frac{\sqrt{3}}{2}\)
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(c)
What is \(\sin30\)°?
  1. \(\displaystyle \frac{\sqrt{3}}{2}\)
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  2. \(\displaystyle \sqrt{3}\)
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  3. \(\displaystyle 2\)
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  4. \(\displaystyle \frac{1}{2}\)
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(d)
What is \(\cos45\)°?
  1. \(\displaystyle \frac{1}{\sqrt{2}}\)
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  2. \(\displaystyle \sqrt{2}\)
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  3. \(\displaystyle 1\)
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  4. \(\displaystyle \frac{\sqrt{2}}{2}\)
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(e)
Now that you have determined \(\tan45\)° (part a), what is \(\cot45\)°?
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Subsection 6.4.2 Exercises

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