Section 2.0 From Trigonometry for High School to Trigonometry
for Calculus
Trigonometry in High School
Encyclopedia Brittanica: "The word trigonometry comes
from the Greek words trigonon (“triangle”) and metron (“to
measure”)."
SOH-CAH-TOA
You might recall this mnemonic for sine, cosine, and tangent of an acute
angle in a right triangle:
SOH—Sine is Opposite over Hypotenuse
CAH—Cosine is Adjacent over Hypotenuse
TOA—Tangent is Opposite over Adjacent
As a pnemonic, SOH-CAH-TOA works well to remind you of what to
divide by what. But it hides the ideas triangle trigonometry is all
about—similarity and proportionality.
The Origins of Trigonometry
Similar Triangles
Ancient Babylonians and Egyptians developed a practical trigonometry
for surveying and map making. They needed to compute distances
between places they could not actually measure. They did this
by drawing a quasi-accurate map of actual locations, then reasoning
proportionally.
For example, Figure 2.0.1 shows a triangle Egyptians drew with side
lengths $s_1$, $s_2$, and $s_3$, which they measured. They drew the
triangle to model a situation involving a much larger actual
triangle for which they could measure only one side ($s'_1$).
Figure 2.0.1 Using proportional reasoning to find
unknown lengths.
The quotient $\dfrac{s_2}{s_1}$ being 0.48 means $s_2$ is 48% as large as
$s_1$. The quotient $\dfrac{s_3}{s_1}$ being 0.65 means $s_3$ is 65% as large
as $s_1$. Since the two triangles are similar, $s'_2$ will be 48% as
large as $s'_1$ and $s'_3$ will be 65% as large as $s'_1$
Reflection 2.0.1. Suppose the larger triangle is made by
three actual villages separated by a river, and $s'_1$ is approximately 29,348
cubits ($1\text{ cubit}\approx 0.523\text{ meters}$). What, approximately, are the other two distances (in cubits)?
Similarity, and thus proportionality, is the heart of high school
trigonometry. Once you know quotients among sides of one triangle,
you know quotients among sides of any triangle similar to it.
Hipparchus, Ptolemy, and Astronomy
Ancient Greeks were fascinated by the stars. They noticed that any star appeared in different positions in the sky when viewed at the same time of night from different locations. They used this observation (now called parallax) as motivation for determining distances from Earth to a star.
We will not delve into their methods for determining distances to stars except to point out they realized that the important aspect of measuring triangles ("trigonometry") was to focus on the measure of an arc on a unit circle in which a right triangle is embedded. They used the length of arc subtended by the angle, measured in units of 1/360 the circle's circumference, as the angle's measure.
The relative sizes among one triangle's sides tells the relative sizes of sides in any triangle similar to it. Hipparchus, and later Ptolemy, used this fact (and geometric theorems from Euclid and Archimedes) to approximate the relative sizes of sides in right triangles with a hypotenuse of length 60 and angles subtending arcs of $\frac{1}{2}^\circ$ and multiples of $\frac{1}{2}^\circ$.
Ptolemy created a table of these values that was used for centuries by anyone needing to reason proportionally about lengths of sides in a right triangle. All they needed to know was the measure of one angle (other than the right angle) in it. They could calculate the (approximate) relative sizes among all three sides of one triangle and then use Ptolemy's table to reason proportionally about any similar triangle's sides based on knowing the length of just one.
Triangles A and B are similar. Triangles C and D are similar. Use relative sizes of sides in Triangle A to determine the approximate lengths of Triangle B's sides. Do the same for Triangles C and D.
Christian Carmen gives a detailed explanation of Ptolemy's calculation of the distance from Earth to Sun. The figure below is adapted from his Figure 1. It shows Earth, Moon, and Sun during a total eclipse of the sun.
Ptolemy determined angle MEH has a measure of approximately 0.26 degree. His table of values said a right triangle with one angle measuring 0.26 degrees has 0.0046 as the quotient of opposite and adjacent lengths. Ptolemy also determined the Sun's radius is approximately 401 times Moon's's radius. According to these approximations, how far away is Sun from Earth (in Moon radii)?
Earth's radius is 3.66 times as large as Moon's radius. According to these approximations, how far is Sun from Earth in Earth radii?
Earth's radius is 3950 miles (6356 km). According to these approximations, how far is Sun from Earth in miles? in kilometers?
What High School Trigonometry Became
Trigonometry for centuries was about triangles centered in a circle made by a chord of a circle, and angle measure historically was about the length of subtended arcs measured in units proportional to the circle's circumference (e.g., 1/360th the circumference). Starting in the early 1900's, textbooks' emphasis moved to ratios of sides in a right triangle and the ideas of circle and arc length eventually disappeared from high school trigonometry.
But Trigonometric Identities Can Be Useful
A trigonometric identity is an equation involving trigonetric functions that is always true.
The most basic identity, called the Pythagorean Identity, is $(\cos A)^2 + (\sin A)^2 = 1$ for any acute angle A in a right triangle. Figure 2.0.2 shows its derivation. It also shows the central role of Pythagoras' Theorem—that in any right triangle with sides of length a and b and hypotenuse c, $a^2+b^2=c^2$.
Figure 2.0.2. An important trigonometric identity based on Pythagoras' Theorem: $(\cos X)^2+(\sin X)^2=1$ for any acute angle X in a right triangle.
Other trigonometric functions are tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). Their definitions, using the triangle in Figure 2.0.2 are:
$$\begin{align}
\tan A &= \frac{a}{b}=\frac{\left(\dfrac{a}{c}\right)}{\left(\dfrac{b}{c}\right)}=\frac{\sin A}{\cos A}\\[1ex]
\cot A &= \frac{b}{a} = \frac {1}{\tan A}\\[1ex]
\csc A &= \frac{c}{a} = \frac {1}{\sin A}\\[1ex]
\sec A &= \frac{b}{a} = \frac {1}{\cos A}
\end{align}$$
Some identities follow directly from the identity $(\cos A)^2+(\sin A)^2=1$. For example:
$$\begin{align}
(\sin A)^2&=1-(\cos A)^2\\[1ex]
(\cos A)^2&=1-(\sin A)^2\\[1ex]
\end{align}$$
More identities can be derived from the ones above. For example:
$$\begin{align}
(\sin A)^2&=1-(\cos A)^2\\[1ex]
\frac
{(\sin A)^2}
{(\cos A)^2} &=
\frac
{1}{(\cos A)^2} - \frac{(\cos A)^2}{(\cos A)^2}\\[1ex]
(\tan A)^2 &= (\sec A)^2 - 1
\end{align}$$
This PDF contains an extensive collection of trigonometric identites. Please note that in this list, θ and x stand for angle measures, not angles themselves. This will make more sense in the next sections.
Exercise Set 2.0.2
Start with the Pythagorean Identity $(\sin A)^2+(\cos A)^2=1$ to end with the identity $(\tan A)^2 + 1 = (\sec A)^2$.
Start with the identity $\sin(A+B)=\sin A\cos B+\sin B\cos A$ to show:
$\sin 2A=2\sin A\cos A$
$\sin 3A=3\sin A-4(\sin A)^3$ Hint: Think of 3A as (2A + A).
$\sin 4A=4\sin A\cos A-8(\sin A)^3\cos A$
Start with the identities $(\sin A)^2+(\cos A)^2=1$ and $\cos (A+B)=\cos A \cos B - \sin A \sin B$ to show:
$\cos 2A = (\cos A)^2 - (\sin A)^2$
$\cos 2A = 1 - 2(\sin A)^2$
$\cos 2A = 2(\cos A)^2 - 1$
Triangle Trigonometry is Important, but Inadequate for Calculus
Calculus is about relationships quantities quantities whose values vary. High school trigonometry is about sides of triangles. Also, the idea of angle measure in high school trigonometry is hardly treated seriously.
It is common to find expressions like $\sin(\angle A)$ or $\sin(A)$ in high school textbooks, where A is a vertex of a triangle. Put another way, sine, cosine, tangent, etc. are portrayed in high school trigonometry as having geometric objects as their arguments. But you cannot put angles on a number line; you can only put angle measures (i.e., numbers) on a number line. Thus, graphs of trigonetric functions (e.g., $y=\sin A$) make no sense when A is an angle or a vertex in a triangle.
The remainder of this chapter will build up the idea of angle measure and will develop trigonometric functions as functions of angle measure.
