Checkpoint 7.1.1.
If the two sides on the right angle have length \(a={1}\) and \(b={2}\text{,}\) what is the length of the hypotenuse? \(c=\)
Answer.
\(2.23607\)
Section 7.1 Trigonometric Ratios
In Section 4.1 we learned about areas of triangles and a relationship between the three sides of a right triangle. In this and the next sections we will look at relationships between angles of the triangles and their sides.
Subsection 7.1.1 Side and Angle Relationships
The Theorem 4.1.16 tells us that if we know two sides of a right triangle, the length of the third side is already determined. This means a right triangle cannot be assembled from three, random side lengths.
Checkpoint 7.1.2.
Can a triangle have one leg (side next to right angle) of length \(a={4}\) and hypotenuse (side opposite the right angle) with length \(c={3}\text{?}\)
- Yes
- No
Answer.
\(\text{No}\)
There is also a relationship between the three angles of any triangle.
Theorem 7.1.3.
The sum of the angles of any triangle is \(180^\circ\text{.}\)
Example 7.1.4.
If two angles of a triangle are \(40^\circ\) and \(70^\circ\) what is the other angle?
Solution.
The third angle must satisfy
\begin{align*}
40+70+\theta & = 180.\\
\theta & = 70.
\end{align*}
Example 7.1.5.
If one angle of a right triangle is \(55^\circ\) what is the other angle?
Solution.
We know one angle is \(90^\circ\) (right angle) and another is \(55^\circ\text{.}\) Thus the third angle must satisfy
\begin{align*}
90+55+\theta & = 180.\\
\theta & = 35.
\end{align*}
Checkpoint 7.1.6.
If two angles of a triangle are 40 and 40, what is the measure of the third angle?
Answer.
\(100\)
Subsection 7.1.2 Defining Trig Functions
Note that in a right triangle the other two angles have measure less than right angles. This is a result of Theorem 7.1.3. Consider that \(180^\circ-90^\circ=90^\circ\) so the remaining two angles have a sum that adds to \(90^\circ\) implying both are smaller.
For right triangles we have names for the sides. Consider the labels in Figure 7.1.7 The adjacent is the side touching the right angle that is also touching the angle with which we are working. The opposite is the side touching the right angle not touching the angle with which we are working. Note these names are relative to the angle we are considering. That is in Figure 7.1.10 the adjacent side for \(\alpha\) has length 5 and the adjacent side for \(\theta\) has length 3. Both the adjacent and opposite are known as legs of the triangle. The hypotenuse is opposite the right angle (the one side not touching it).
Similar to this restriction on the lengths of the three sides, there are restrictions on the ratio of side lengths given the measure of the angles. Use the activity in Figure 7.1.8 to see how changing either side length or angle affects the other.
Activity 9.
The following steps show us that it makes sense to define ratios of side lengths of right triangles. Use the activity in Figure 7.1.8
(a)
Use the slider for \(\alpha\) to increase the angle from \(0^\circ\) to \(90^\circ\text{.}\) As the angle increases what does the length of the opposite side (j) do?
(b)
Use the slider for \(\alpha\) to increase the angle from \(0^\circ\) to \(90^\circ\text{.}\) As the angle increases what does the length of the adjacent side (j) do?
(c)
Note that the hypotenuse does not change in this example. Based on your result in Task 9.a, as the angle \(\alpha\) increases what will the ratio of opposite to hypotenuse (j/g) do?
(d)
Note that the hypotenuse does not change in this example. Based on your result in Task 9.b, as the angle \(\alpha\) increases what will the ratio of adjacent to hypotenuse (i/g) do?
(e)
Instructions.
Use the slider to adjust the angle and see how the side lengths and their ratio change.
Because the ratios are dependent solely on the angle it is reasonable to name and use them. The trigonometric functions (names for the ratios) are in Table 7.1.9
| sine | \(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\) |
| cosine | \(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\) |
| tangent | \(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\) |
| secant | \(\sec(\alpha) = \frac{\text{hypotenuse}}{\text{adjacent}}\) |
| cosecant | \(\csc(\alpha) = \frac{\text{hypotenuse}}{\text{opposite}}\) |
| cotangent | \(\cot(\alpha) = \frac{\text{adjacent}}{\text{opposite}}\) |
Example 7.1.11.
Given the side lengths in Figure 7.1.10 what are each of the following trig ratios?
(a)
\(\sin(\alpha)=\)
Solution.
From the perspective of \(\alpha\text{,}\) the side of length 3 is opposite. The hypotenuse has length \(\sqrt{34}\text{.}\) Thus
\begin{equation*}
\sin(\alpha) = \frac{3}{\sqrt{34}}
\end{equation*}
(b)
\(\cos(\alpha)=\)
Solution.
From the perspective of \(\alpha\text{,}\) the side of length 5 is adjacent. The hypotenuse has length \(\sqrt{34}\text{.}\) Thus
\begin{equation*}
\cos(\alpha) = \frac{5}{\sqrt{34}}
\end{equation*}
(c)
\(\sec(\alpha)=\)
Solution.
\(\sec(\alpha)\) flips the ratio of \(\cos(\alpha)=\frac{5}{\sqrt{34}}\) (from the previous problem). Thus
\begin{equation*}
\sec(\alpha) = \frac{\sqrt{34}}{3}
\end{equation*}
(d)
\(\sin(\theta)=\)
Solution.
From the perspective of \(\theta\text{,}\) the side of length 5 is opposite. The hypotenuse has length \(\sqrt{34}\text{.}\) Thus
\begin{equation*}
\sin(\theta) = \frac{5}{\sqrt{34}}
\end{equation*}
Defining the trigonometric functions via ratios has an inherent limitation.
Example 7.1.12.
Consider a right triangle with side lengths 8, 15, and 17. If \(A\) is the angle across from 8, then \(\sin(A)=\frac{8}{17}\text{.}\)
Next consider a right triangle with side lengths 16, 30, and 34. If \(A\) is the angle across from 16, then \(\sin(A)=\frac{16}{34}=\frac{8}{17}\text{.}\) This is the same ratio as the previous triangle although the triangle is larger (double in each side length).
Trigonometric functions are ratios which implies that they do not contain information about scale.
It is also possible to find the angle given the ratio. We use the so called inverse trigonometric functions for this. There are two common notations for them which are shown in Table 7.1.13.
| Trig | Inverse Trig | |
| \(\sin \alpha = r\) | \(\arcsin r = \alpha\) | \(\sin^{-1} r = \alpha\) |
| \(\cos \alpha = r\) | \(\arccos r = \alpha\) | \(\cos^{-1} r = \alpha\) |
| \(\tan \alpha = r\) | \(\arctan r = \alpha\) | \(\tan^{-1} r = \alpha\) |
Note the notation \(\sin^{-1} x\) shows up on calculator keys and in many books. It is unfortunately easy to confuse with \((\sin x)^{-1} = \frac{1}{\sin x} = \csc x\text{.}\) The relationship between \(\csc x\) and \(\sin x\) will not be used in this chapter. Be careful to read the full context of any problem. Your calculator does not have a button for \((\sin x)^{-1}\) and probably not for \(\csc x\text{.}\) For those you use the \(1/x\) button, that is, calculate the \(\sin x\) first then invert.
Example 7.1.14.
What is the measure of both non-right angles in Figure 7.1.10?
Solution 1.
We can use the arcsine function.
\begin{equation*}
\alpha = \arcsin(3/\sqrt{34}) \approx 31^\circ\text{.}
\end{equation*}
\begin{equation*}
\theta = \arcsin(5/\sqrt{34}) \approx 59^\circ
\end{equation*}
Solution 2.
We can use the arccosine function.
\begin{equation*}
\alpha = \arccos(5/\sqrt{34}) \approx 31^\circ\text{.}
\end{equation*}
\begin{equation*}
\theta = \arccos(3/\sqrt{34}) \approx 59^\circ
\end{equation*}
Example 7.1.15.
A right triangle has legs of lengths 4 and 8. What are the measures of the non-right angles?
Solution 1.
Because we have the two legs, we can use the arctangent function to calculate the angles.
\begin{align*}
\arctan\left(\frac{4}{8}\right) \amp \approx 26.57.\\
\arctan\left(\frac{8}{4}\right) \amp \approx 63.43.
\end{align*}
Solution 2.
Because we have two legs, we can use the Pythagorean Theorem to calculate the third side length, then use arcsine.
\begin{align*}
4^2+8^2 \amp = c^2.\\
80 \amp = c^2.\\
8.944 \amp \approx c.
\end{align*}
\begin{align*}
\arctan\left(\frac{4}{8.944}\right) \amp \approx 26.57.\\
\arctan\left(\frac{8}{8.944}\right) \amp \approx 63.44.
\end{align*}
Notice that the larger angle is slightly different from the first solution. This is the result of using the approximate hypotenuse.
Checkpoint 7.1.16.
If the leg lengths of a right triangle are 5 and 8, what are the measures of the angles?
Angle opposite side length 5:
Angle opposite side length 8:
Answer 1.
\(\tan^{-1}\!\left(\frac{5}{8}\right)\)
Answer 2.
\(\tan^{-1}\!\left(\frac{8}{5}\right)\)
Subsection 7.1.3 Solving Triangles
Our goal now is to use partial information about a triangle to find the rest.
Example 7.1.18.
If \(\sin \alpha = \frac{12}{13}\) and the hypotenuse has length 13, what is the length of the adjacent side?
Solution.
Sine is opposite over hypotenuse. Because we know the hypotenuse is 13 this ratio tells us the opposite is 12. Now we are looking for the length of the adjacent.
\begin{align*}
12^2+b^2 & = 13^2.\\
b^2 & = 13^2-12^2.\\
b^2 & = 25.\\
b & = 5.
\end{align*}
Example 7.1.19.
If \(\sin \alpha = 0.2800\) and the opposite side for \(\alpha\) has length 14.00, what are the lengths of the other sides?
Solution.
Sine is opposite over hypotenuse so, we can setup a proportion.
\begin{align*}
\frac{14.00}{h} \amp = 0.2800.\\
14.00 \amp = h \cdot 0.2800. \amp \text{ clearing the denominator}\\
\frac{14.00}{0.2800} \amp = h.\\
50.00 \amp = h.
\end{align*}
The hypotenuse has length 50.00.
\begin{align*}
14.00^2+b^2 \amp = 50.00^2.\\
b^2 \amp = 50.00^2-14.00^2.\\
b^2 \amp = 2304.\\
\sqrt{b^2} \amp = \sqrt{2304}.\\
b \amp = 48.00.
\end{align*}
The adjacent has length 48.00.
Checkpoint 7.1.20.
For a right triangle if \(\sin(\alpha)=\frac{{5}}{{12.083}}\text{,}\) what is the length of the adjacent?
Answer.
\(11\)
Example 7.1.21.
For a right triangle if \(\sin \alpha = \frac{12}{13}\text{,}\) what are the two, non-right angles?
Solution.
We can use \(\arcsin(12/13)\) for angle \(\alpha\text{.}\) This is \(\arcsin(12/13) \approx 67^\circ\text{.}\) For the second angle we have three options. First we can use the angle sum.
\begin{align*}
90+67+\theta & = 180.\\
\theta & = 23.
\end{align*}
Another option is to recognize that 12/13 is adjacent over hypotenuse for the third angle. Thus it is given by \(\arccos(12/13) \approx 23\text{.}\)
It is also possible to use the third side. We know from Example 7.1.18 that the third side length is 5. Thus the angle is given by \(\arcsin(5/13) \approx 23\text{.}\)
We can select a favorite method in cases like these.
Example 7.1.22.
For a right triangle with angle \(\alpha=50^\circ\) and opposite side length 7, what are the other side lengths and angles? All numbers given are exact.
Solution.
Because sine is opposite over hypotenuse and we know both the angle and opposite, we can calculate the hypotenuse.
\begin{align*}
\sin(50^\circ) \amp \approx 0.766.\\
\frac{7}{h} \amp \approx 0.766.\\
7 \amp \approx h \cdot 0.766. \amp \text{ clearing the denominator}\\
\frac{7}{0.766} \amp \approx h.\\
9.13 \amp \approx h.
\end{align*}
The hypotenuse has length 9.13. Next we calculate the length of the adjacent.
\begin{align*}
7^2+b^2 \amp = 9.13^2.\\
b^2 \amp = 9.13^2-7^2.\\
b^2 \amp = 34.3569.\\
\sqrt{b^2} \amp = \sqrt{34.3569}.\\
b \amp \approx 5.86.
\end{align*}
The adjacent has length 5.86. Finally, we know that two of the angles are \(90^\circ\) and \(50^\circ\text{,}\) so the third angle has measure \(180^\circ-90^\circ-50^\circ = 40^\circ\text{.}\)
Checkpoint 7.1.23.
If a triangle has a leg of length 6 and hypotenuse of length 11, what is the length of the other side and what are the measures of the angles?
Length of other leg:
Angle opposite side length 6:
Angle opposite other leg:
Answer 1.
\(9.21954\)
Answer 2.
\(\sin^{-1}\!\left(\frac{6}{11}\right)\)
Answer 3.
\(\cos^{-1}\!\left(\frac{6}{11}\right)\)
Exercises 7.1.4 Exercises
Exercise Group.
Use the Pythagorean Theorem and angle sum fact to calculate side lengths and angles.
1. Triangle Side Length.
2. Triangle Side Length.
3. Triangle Angles.
4. Triangle Angles.
Exercise Group.
Use the ratio definitions of trigonometric functions to answer these.
5. Right Triangle Side Names.
6. Trig Function Definitions.
7. Trig Function Value.
8. Trig Function Value.
Exercise Group.
Calculate side lengths and angles using trigonometric functions.
