6.5
Trigonometry
Suppose you wanted to measure the height of a tree. You could wriggle along the ground away from the base of the tree until, when you sighted along the diagonal of a square, the sight line went right to the top of the tree. You, the base of the tree, and the top of the tree form a 45–45–90 right triangle, so the height of the tree equals the distance from you to the base of the tree. You've solved the problem, but surely there's an easier way to get the result.
We can then set up a simple arithmetic problem: $$0.5543=\frac{AB}{AB}=\frac{BC}{100},$$ giving $BC=55.43$.
It turns out that there were bigger fish to fry, that there were compelling reasons to create such a table of ratios. Imagine that you had never been told which was farther away, the sun or the moon. Imagine that you were living 4000 years ago, and no one knew which was farther away, and you wanted to answer the question. Could you do it? You might say that since it is cooler when the moon is up, the moon must be farther away. Or you might say that since the moon eclipses the sun and not the other way around is clear proof that the moon is closer. That seems compelling. But the next question is: how much closer? The question seems overwhelmingly difficult but you and countless others have seen the event that reveals the answer. During the day it is often the case that the moon is up. More rarely, but still often you can see the sun and a half moon. The diagram below shows how the earth, sun, and moon are oriented in that case.
The word trigonometry comes from Greek words that mean “triangle measure.” In this section we will explore the three basic trigonometric ratios of sine, cosine, and tangent. Like many of the ideas in this chapter, the development of these ratios will be based on the Pythagorean Theorem and similar triangles.
Consider the following 30–60–90 triangles.
\begin{align}
\text{In }\triangle ABC, \frac{b}{c}=\frac{1}{2}\qquad&\text{In }\triangle{FGH}, \frac{g}{h}=\frac{\frac{3}{2}}{3}=\frac{1}{2}\qquad&\text{In }\triangle{IJK}, \frac{j}{k}=\frac{2}{4}=\frac{1}{2}
\end{align}
You should also verify that the ratio of the side adjacent to the $30^\circ$ angle to the hypotenuse is the same for each triangle. In particular, $$\frac{{\text{leg adjacent to }30^\circ}}{\text{hyptenuse}}=\frac{\sqrt{3}}{2}.$$
In fact, the ratio of the leg opposite the $30^\circ$ angle to the leg adjacent to the $30^\circ$ angle is also the same for all three triangles. $$\frac{\text{leg opposite the }30^\circ \angle}{\text{leg adjacent to }30^\circ \angle}=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$$
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A graphing calculator can quickly find the trigonometric ratio of any angle to great accuracy. Locate the SIN, COS, and TAN buttons on your calculator. For example, with your calculator in DEG mode, key in TAN 55 ENTER to get $$\tan 55^\circ=1.4281480007.$$
This link to a Table of Trigonometric Ratios produces the same results.
Example: $\triangle PQR$ has a right angle at $P$, and $m\angle Q=63$ while $PR=7.2$. Determine the length of the hypotenuse $\overline{QR}$. |
The real power of the trigonometric ratios is the fact that given the sides of any right triangle, you can determine the measure of the angles.
Example: Determine the measure
of $\angle A$ in the 3–4–5 right triangle pictured. |
A graphing calculator can increase the accuracy of this estimate. Locate the $\fbox{TAN$^{–1}$}$ button on your calculator. (It is actually the 2nd feature of the $\fbox{TAN}$ button.) With your calculator in DEG mode, key in $\fbox{2ND}$ $\fbox{TAN}$ $\fbox{ . }$ $\fbox{ 7 }$ $\fbox{ 5 }$ to get $$\tan^{-1}.75=36.86.89765^\circ.$$
It is critically important that these ratios be thought of as applicable in right triangles only. If a given triangle does not have a right angle, an altitude must be drawn in order to set up right triangles in which these trigonometric ratios are applicable.
The first Table of Trigonometric Ratios was created by Hipparchus of Nicaea (ca. 180-125 BCE), but the first excellent Table of Trigonometric Ratios was created by Ptolemy of Alexandria who was born at the end of the first century AD. In developing such a table, it should be noted that the values should be as accurate as possible. It is not possible to be very accurate by drawing triangles and measuring them. The table must be created theoretically and ideally. This is where special right triangles played a crucial role. The Table of Trigonometric Ratios says that $\sin 60^\circ=0.8660$, but it is actually $\frac{\sqrt{3}}{2}$. In fact, almost all of the values in the table are approximations of irrational numbers. From problems 21 and 22 of the last section, we could find the exact trigonometric values for $18^\circ$, $72^\circ$, $36^\circ$, and $54^\circ$ angles. Using the Triangle Angle Bisector Theorem, we could use a 30–60–90 triangle to find the exact values for a $15^\circ$ angle, a $7.5^\circ$ angle, and so on. Using the same theorem, we could start with a 18–72–90 triangle and obtain exact values for a $9^\circ$ angle, a $4.5^\circ$ angle, and so on. This would be a tedious process, but once done, it would be done for eternity. Few things that we do last as long. Once done, we have a magnificent tool for indirect measurement, whether it be trees, buildings, or the heavens.