Definition 2.11. Sine of an Acute Angle.
[latex]\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}[/latex]
Chapter 2: Trigonometric Ratios
2.2 Right Triangle Trigonometry
Algebra Refresher
Algebra Refresher
Write two more ratios equivalent to the given fraction.
1. [latex]\:\dfrac{10}{4}[/latex]
2. [latex]\:\dfrac{6}{8}[/latex]
3. [latex]\:0.6[/latex]
4. [latex]\:1.5[/latex]
Compute the slope of the line.
5.
6.
7.
8.
Solve.
9. [latex]\:\dfrac{12}{x} = 48[/latex]
10. [latex]\:\dfrac{60}{x} = 80[/latex]
Algebra Refresher Answers
(Many answers are possible for 1–4.)
- [latex]\displaystyle \dfrac{5}{2},~~\dfrac{20}{8}[/latex]
- [latex]\displaystyle \dfrac{3}{4},~~\dfrac{12}{16}[/latex]
- [latex]\displaystyle \dfrac{3}{5},~~\dfrac{12}{20}[/latex]
- [latex]\displaystyle \dfrac{6}{4},~~\dfrac{12}{8}[/latex]
- [latex]\displaystyle \dfrac{2}{5}[/latex]
- [latex]\displaystyle \dfrac{-8}{5}[/latex]
- [latex]\displaystyle \dfrac{-6}{5}[/latex]
- [latex]\displaystyle \dfrac{1}{2}[/latex]
- [latex]\displaystyle \dfrac{1}{4}[/latex]
- [latex]\displaystyle \dfrac{3}{4}[/latex]
Learning Objectives
- Use measurements to calculate the trigonometric ratios for acute angles
- Use trigonometric ratios to find unknown sides of right triangles
- Solve problems using trigonometric ratios
- Use trig ratios to write equations relating the sides of a right triangle
- Use relationships among the trigonometric ratios
With the Pythagorean theorem we can find one side of a right triangle if we know the other two sides. By using what we know about similar triangles, we can find the unknown sides of a right triangle if we know only one side and one of the acute angles.
The Sine of an Angle
In Example 2 of Section 1.2, we saw that in a 30-60-90 right triangle, the ratio of the shortest side to the hypotenuse was [latex]\frac{1}{2}{,}[/latex] or [latex]0.5[/latex]. This ratio is the same for any two right triangles with a [latex]30°[/latex] angle because they are similar triangles, as shown at right.
The ratio is given a name; it is called the “sine of [latex]30°[/latex]“. We write
[latex]\sin 30° = 0.5,[/latex]
where sin is an abbreviation for sine. There is nothing special about [latex]30°[/latex] angles; we can talk about the sine of any angle. The sine of an angle is the ratio of the side opposite the angle to the hypotenuse.
Example 2.12.
Example 2.12.
Find the sine of the labeled angle in each triangle below.
Solution
-
-
- The side opposite angle [latex]\phi[/latex] is [latex]5[/latex], and the hypotenuse is [latex]13[/latex], so the sine of [latex]\phi[/latex] is
[latex]\sin \phi =\dfrac{5}{13}, \text{ or approximately} \ 0.3846[/latex] - The side opposite angle [latex]\beta[/latex] is [latex]\sqrt{5}{,}[/latex] and the hypotenuse is [latex]3[/latex], so the sine of [latex]\beta[/latex] is
[latex]\sin \beta =\dfrac{\sqrt{5}}{3}, \text{ or approximately} \ 0.7454[/latex]
- The side opposite angle [latex]\phi[/latex] is [latex]5[/latex], and the hypotenuse is [latex]13[/latex], so the sine of [latex]\phi[/latex] is
-
Checkpoint 2.13.
Find the sine of the labeled angle in the triangle at right. Round your answer to 4 decimal places.
Solution
[latex]0.7442[/latex]
Caution 2.14.
We must use the sides of a right triangle to calculate the sine of an angle. For example, in the triangle at right, [latex]\sin \theta=\frac{4}{7}{,}[/latex] because [latex]\triangle ABC[/latex] is a right triangle. It is not true that [latex]\sin \theta=\frac{5}{7}{,}[/latex] or [latex]\sin \theta=\frac{6}{7}{.}[/latex] In this chapter, we consider only right triangles.
Using a Calculator
Mathematicians have calculated the sines of any angle we like. The values of the sine were originally collected into tables and are available on scientific calculators. For example, let’s find the sine of [latex]50°{.}[/latex] First, consider some triangles, as shown below.
[latex]{\text{Which angle has the larger sine, } 30°\ \text{or}\ 50°?}[/latex]
Do you expect the sine of [latex]50°[/latex] to be larger or smaller than the sine of [latex]30°\text{?}[/latex] Do you expect the sine of [latex]50°[/latex] to be larger or smaller than 1?
Example 2.15.
Use your calculator to find the sine of [latex]50°[/latex] by entering SIN [latex]50{.}[/latex] (Make sure your calculator is set for degrees.) You should find that
[latex]\sin 50° = 0.7660444431\text{.}[/latex]
This is not an exact value; the sine of [latex]50°[/latex] is an irrational number, and your calculator shows as many digits as its display will allow. (Not all sine values are as “nice” as the sine of [latex]30°\text{!}[/latex]) Usually we round to four decimal places, so we write
[latex]\sin 50° = 0.7660[/latex]
[latex]{\sin} ([/latex]
with an open parenthesis, as the prompt to enter an angle. It is more proper to use parentheses and write sin [latex](50°)[/latex] for “the sine of [latex]50°{,}[/latex]” but we often omit the parentheses in simple trigonometric expressions. Think of the notation sin as an operation symbol telling you to find the sine of an angle, just as the symbol [latex]\sqrt{}[/latex] tells you to take the square root of the expression under the radical.
Checkpoint 2.17.
- Use your calculator to complete the table, rounding your answers to four decimal places.
[latex]\theta[/latex] [latex]0 °[/latex] [latex]10 °[/latex] [latex]20 °[/latex] [latex]30 °[/latex] [latex]40 °[/latex] [latex]50 °[/latex] [latex]60 °[/latex] [latex]70 °[/latex] [latex]80 °[/latex] [latex]90 °[/latex] [latex]\sin \theta[/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] - What do you notice about the values of [latex]\sin \theta[/latex] as [latex]\theta[/latex] increases from [latex]0 °[/latex] to [latex]90 °\text{?}[/latex] If you plot the values of [latex]\sin \theta[/latex] against the values of [latex]\theta{,}[/latex] will the graph be a straight line? Why or why not?
Solution
-
[latex]\small \theta[/latex] [latex]\small 0 °[/latex] [latex]\small 10 °[/latex] [latex]\small 20 °[/latex] [latex]\small 30 °[/latex] [latex]\small 40 °[/latex] [latex]\small 50 °[/latex] [latex]\small 60 °[/latex] [latex]\small 70 °[/latex] [latex]\small 0 °[/latex] [latex]\small 90 °[/latex] [latex]\small \sin \theta[/latex] [latex]~~0[/latex] [latex]\small 0.1737[/latex] [latex]\small 0.3420[/latex] [latex]\small 0.5[/latex] [latex]\small 0.6428[/latex] [latex]\small 0.7660[/latex] [latex]\small 0.8660[/latex] [latex]\small 0.9397[/latex] [latex]\small 0.9848[/latex] [latex]~~1[/latex] - The values of [latex]\sin \theta[/latex] increase from 0 to 1 as [latex]\theta[/latex] increases from [latex]0°[/latex] to [latex]90°\text{.}[/latex] The graph will not be a straight line because the slopes between successive points are not constant.
Note 2.18.
The important thing to remember is that the sine of an angle, say [latex]50 °\text{,}[/latex] is the same for any right triangle with a [latex]50 °[/latex] angle, no matter what the size or orientation of the triangle.
The figure below shows three different right triangles with a [latex]50°[/latex] angle. Although the sides of the triangle may be bigger or smaller, the ratio [latex]\dfrac {\text{opposite}}{\text{hypotenuse}}[/latex] is always the same for that angle, because the triangles are similar. This is why the sine ratio is useful.
[latex]{\text{In each triangle, the ratio}~ \sin 50°=\dfrac{\text{opposite}}{\text{hypotenuse}}=0.7660}[/latex]
Using the Sine Ratio to Find an Unknown Side
In the next example, we see how to use the sine ratio to find an unknown side in a right triangle, knowing only one other side and one angle.
Example 2.19.
Example 2.19.
Find the length of the side opposite the [latex]50°[/latex] angle in the triangle shown.
Solution
In this triangle, the ratio [latex]\dfrac {\text{opposite}}{\text{hypotenuse}}[/latex] is equal to the sine of [latex]50 °{,}[/latex] or
[latex]\sin 50° = \dfrac {\text{opposite}}{\text{hypotenuse}}[/latex]
We use a calculator to find an approximate value for the sine of [latex]50°{,}[/latex] filling in the lengths of the hypotenuse and the opposite side to get
[latex]0.7660 = \dfrac {x}{18}[/latex]
We solve for [latex]x[/latex] to find
[latex]x = 18(0.7660)=13.788[/latex]
To two decimal places, the length of the opposite side is [latex]13.79[/latex] centimeters.
Caution 2.20.
In the previous example, even though we showed only four places in [latex]\sin 50°{,}[/latex] you should not round off intermediate steps in a calculation, because the answer loses accuracy with each rounding. You can use the following keystrokes on your calculator to avoid entering a long approximation for [latex]\sin 50°{:}[/latex]
[latex]{sin}~(50) \times 18[/latex]
The calculator returns [latex]x=13.78879998.[/latex]
Checkpoint 2.21.
Find the length of the hypotenuse in the triangle shown.
Solution
8.5 m
The Cosine and the Tangent
There are two more trigonometric ratios used for calculating the sides of right triangles, depending on which of the three sides is known and which are unknown. These ratios are called the cosine and the tangent.
Suppose we’d like to find the height of a tall cliff without actually climbing it. We can measure the distance to the base of the cliff, and we can use a surveying tool called a theodolite to measure the angle between the ground and our line of sight to the top of the cliff (this is called the angle of elevation).
These values give us two parts of a right triangle, as shown at right. The height we want is the side opposite the angle of elevation. The distance to the base of the cliff is the length of the side adjacent to the angle of elevation.
The ratio of the side opposite an angle to the side adjacent to the angle is called the tangent of the angle. The abbreviation for “tangent of theta” is tan [latex]\theta[/latex].
Definition 2.22. Tangent of an Acute Angle.
[latex]\qquad\tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}[/latex]
Just like the sine of an angle, the tangent ratio is always the same for a given angle, no matter what size triangle it occurs in. And just like [latex]\sin \theta{,}[/latex] we can find the values of [latex]\tan \theta[/latex] on a scientific calculator.
Example 2.23.
- Use your calculator to find the tangent of [latex]58°{.}[/latex]
- Find the height of the cliff if the angle of elevation to the top of the cliff is [latex]58°[/latex] at a distance of [latex]300[/latex] feet from the base of the cliff.
Solution
- Enter TAN 58 to find [latex]\tan 58° = 1.6003{,}[/latex] rounded to four decimal places.
- We use the tangent ratio to write an equation. In this triangle, the angle is [latex]58°{.}[/latex]
[latex]\tan 58° = \dfrac{{opposite}}{{adjacent}}[/latex]
Next we fill in the value of [latex]\tan 58°{,}[/latex] and the lengths of the sides.
[latex]1.6003 = \frac{h}{300}[/latex]
Solving for [latex]h[/latex] gives
[latex]h = 300(1,6003) = 480.0[/latex]
so the height of the cliff is about [latex]480[/latex] feet.
Checkpoint 2.24.
- Use the tangent ratio to find [latex]x[/latex] in the triangle shown.
- Use the sine ratio to find the hypotenuse, [latex]c{,}[/latex] of the triangle.
- Use the Pythagorean theorem to find the hypotenuse of the triangle. Do you get the same answer with both methods? Can you explain why the calculations might give (slightly) different answers?
Solution
- [latex]x = 23[/latex] ft
- [latex]c = 55[/latex] ft
- The answers agree when rounded to units. Rounding during calculation can cause the results to differ.
The third trigonometric ratio, called the cosine, is the ratio of the side adjacent to an angle and the hypotenuse of the triangle.
Definition 2.25. Cosine of an Acute Angle.
[latex]\cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}[/latex]
Example 2.26.
Find [latex]\sin \theta,~ \cos \theta{,}[/latex] and [latex]\tan \theta[/latex] for the triangle shown at right.
Solution
First, we use the Pythagorean theorem to find the hypotenuse, [latex]c{.}[/latex]
[latex]c^2 =6^2 + 8^2\\ c^2 = 536=84 = 100 {{Take square roots.}}\\ c = \sqrt{100} = 10[/latex]
For the angle [latex]\theta{,}[/latex] the opposite side is 8 inches long, and the adjacent side is 6 inches long, as shown in the figure. Thus,
[latex]\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{10}~~ {or}~~ 0.8\\ \cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}} = \frac{6}{10} ~~ {or}~~ 0.6\\ \tan \theta = \dfrac{\text{opposite}}{\text{adjacent}} = \frac{8}{6} ~~ {or}~~1.\overline{3}[/latex]
Checkpoint 2.27.
- Use your calculator to complete the table. Rounding the values of sine and cosine to four decimal places.
[latex]\theta[/latex] [latex]0 °[/latex] [latex]10 °[/latex] [latex]20 °[/latex] [latex]30 °[/latex] [latex]40 °[/latex] [latex]50 °[/latex] [latex]60 °[/latex] [latex]70 °[/latex] [latex]80 °[/latex] [latex]90 °[/latex] [latex]\sin \theta[/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex]\cos \theta[/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] [latex][/latex] - What do you notice about the values of sine and cosine? Can you explain why this is true? (Hint: If one [non-right] angle of a right triangle measures [latex]x[/latex] degrees, how big is the other angle? Now sketch the triangle and label the opposite and adjacent sides for each angle.)
Solution
-
[latex]\small \theta[/latex] [latex]\small 0 °[/latex] [latex]\small 10 °[/latex] [latex]\small 20 °[/latex] [latex]\small 30 °[/latex] [latex]\small 40 °[/latex] [latex]\small 50 °[/latex] [latex]\small 60 °[/latex] [latex]\small 70 °[/latex] [latex]\small 80 °[/latex] [latex]\small 90 °[/latex] [latex]\small \sin \theta[/latex] [latex]\small 0[/latex] [latex]\small 0.1737[/latex] [latex]\small 0.3420[/latex] [latex]\small 0.5[/latex] [latex]\small 0.6428[/latex] [latex]\small 0.7660[/latex] [latex]\small 0.8660[/latex] [latex]\small 0.9397[/latex] [latex]\small 0.9848[/latex] [latex]\small 1[/latex] [latex]\small \cos \theta[/latex] [latex]\small 1[/latex] [latex]\small 0.9848[/latex] [latex]\small 0.9397[/latex] [latex]\small 0.8660[/latex] [latex]\small 0.7660[/latex] [latex]\small 0.6428[/latex] [latex]\small 0.5[/latex] [latex]\small 0.3420[/latex] [latex]\small 0.1737[/latex] [latex]\small 0[/latex] - The cosine of [latex]\theta[/latex] is equal to the sine of the complement of [latex]\theta{,}[/latex] or [latex]\cos \theta = \sin (90-\theta){.}[/latex]
Note 2.28.
In the previous exercise, you should also notice that as the angle [latex]\theta[/latex] increases, [latex]\sin \theta[/latex] increases but [latex]\cos \theta[/latex] decreases.
You can see why this is true in the figure below. In each right triangle, the hypotenuse has the same length. But as the angle increases, the opposite side gets longer and the adjacent side gets shorter.
The Three Trigonometric Ratios
Here is a summary of the three trigonometric ratios we have discussed.
Trigonometric Ratios.
If [latex]\theta[/latex] is one of the angles in a right triangle,
[latex]\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}\\ \cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}\\ \tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}[/latex]
These three definitions are the foundation for all the rest of trigonometry. You must memorize them immediately!!
You must also be careful to apply these definitions of the trigonometric ratios only to right triangles. In the next example, we create a right triangle by drawing an extra line.
Example 2.29.
Example 2.29.
The vertex angle of an isosceles triangle is [latex]34°{,}[/latex] and the equal sides are 16 meters long. Find the altitude of the triangle.
Solution
The triangle described is not a right triangle. However, the altitude of an isosceles triangle bisects the vertex angle and divides the triangle into two congruent right triangles, as shown in the figure. The [latex]16[/latex]-meter side becomes the hypotenuse of the right triangle, and the altitude, [latex]h{,}[/latex] of original triangle is the side adjacent to the [latex]17°[/latex] angle.
Which of the three trig ratios is helpful in this problem? The cosine is the ratio that relates the hypotenuse and the adjacent side, so we’ll begin with the equation
[latex]\cos 17° =\dfrac{{adjacent}}{{hypotenuse}}[/latex]
We use a calculator to find [latex]\cos 17°[/latex] and fill in the lengths of the sides.
[latex]0.9563 = \dfrac {h}{16}[/latex]
Solving for [latex]h[/latex] gives
[latex]h = 16(0.9563) = 15.3008[/latex]
The altitude of the triangle is about [latex]15.3[/latex] meters long.
Checkpoint 2.30.
Another isosceles triangle has base angles of [latex]72°[/latex] and equal sides of length 6.8 centimeters. Find the length of the base.
Solution
4.2 cm
Section 2.2 Summary
Vocabulary
Concepts
- By using similar triangles, we can find the unknown sides of a right triangle if we know only one side and one of the acute angles.
-
Trigonometric Ratios.
If [latex]\theta[/latex] is one of the angles in a right triangle,
[latex]\sin \theta = \dfrac{\text{opposite}}{\text{hypotenuse}}\\ \cos \theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}\\ \tan \theta = \dfrac{\text{opposite}}{\text{adjacent}}[/latex]
- The trigonometric ratio of an angle [latex]\theta[/latex] is the same for every right triangle containing the angle.
Study Questions
- Sketch a figure that illustrates why [latex]\cos 25°[/latex] is the same for every right triangle with a [latex]25°[/latex] angle.
- Sketch a figure that illustrates why [latex]\cos \theta[/latex] decreases as [latex]\theta[/latex] increases from [latex]0°[/latex] to [latex]90°{.}[/latex]
- Which trigonometric ratio would you use to find the hypotenuse of a right triangle if you knew one acute angle and the side opposite that angle?
- Does your calculator give you the exact decimal values for the trigonometric ratios of acute angles?
definition
Sine is a trigonometric function equal to the ratio of the length of the side opposite a given angle in a right triangle to the length of the hypotenuse, expressed as \[ \sin ( \theta )= \frac{\text{opposite} }{\text{hypotenuse}}. \]
Cosine is a trigonometric function equal to the ratio of the length of the adjacent side a given angle in a right triangle to the length of the hypotenuse, expressed as \[ cos ( theta = \frac{\text{opposite}}{\text{hypotenuse}} \].
Tangent is a trigonometric function that describes the relationship between the sides of a right triangle. It is equal to the ratio of the length of the side opposite a given angle to the length of the adjacent side, expressed as tan(θ) = opposite/adjacent. In other words, the tangent of an angle in a right triangle is equal to the length of the side opposite the angle divided by the length of the adjacent side.
The elevation angle is the angle between the horizontal plane and the line of sight from an observer to an object or point in space.
In trigonometry, the adjacent side is the side of a right triangle that is adjacent, or next to, a given angle. It is the side that forms one of the two acute angles in the right triangle, along with the hypotenuse and the opposite side. The adjacent side is always the side that is adjacent to the angle for which we are calculating a trigonometric function such as cosine or tangent
An irrational number is a real number that cannot be expressed as a ratio of two integers, and its decimal expansion is non-repeating and non-terminating.