Calculus Morris Kline Inclination Of A Line

[speedlearner]

Homework Statement

On pages 65-66 of Calculus An Intuitive And Physical Approach, Mr. Kline discusses Inclinations of lines. In illustration 3A-7 he uses the substitution method to create a proof. But there is one step that I think he leaves out. He shows the equation tan A = tan(180 - B) = -tan(B). I understand that angle A = (180 - B) So I understand the first part of the equation. But what I want to know is what he does to the number 180 in the second part of the equation.

Homework Equations

tan(A) = tan(180 - B) = -tan(B)

The Attempt at a Solution

After researching Ptolomey's table of Chords and Rene Descartes, I know that negative numbers were not used in early mathematics. I also know that Euclid himself never considered negative numbers. I've heard that Fermat and Issac Newton were some of the first people to use negative numbers. But this still doesn't completely explain what happens to the number 180 in the second part of the equation.

 

[Mark44]

Homework Statement

On pages 65-66 of Calculus An Intuitive And Physical Approach, Mr. Kline discusses Inclinations of lines. In illustration 3A-7 he uses the substitution method to create a proof. But there is one step that I think he leaves out. He shows the equation tan A = tan(180 - B) = -tan(B). I understand that angle A = (180 - B) So I understand the first part of the equation. But what I want to know is what he does to the number 180 in the second part of the equation.

I don't have that textbook, so I assume that Kline also assumes that A and B are supplementary angles; i.e., that they add to 180°.

Homework Equations

tan(A) = tan(180 - B) = -tan(B)

The Attempt at a Solution

After researching Ptolomey's table of Chords and Rene Descartes, I know that negative numbers were not used in early mathematics. I also know that Euclid himself never considered negative numbers. I've heard that Fermat and Issac Newton were some of the first people to use negative numbers. But this still doesn't completely explain what happens to the number 180 in the second part of the equation.

It's really 180°, the number of degrees for an angle whose two rays form a straight line. Using the definition of the tangent as the rise/run for an acute angle of a right triangle, it's easy to see that the reference triangle for angle A and the reference triangle for angle B are congruent. This means that the rises of the two triangles are equal, and the bases are of the same length but pointing in the opposite directions. Therefore, tan(A) = -tan(B).

There's really nothing mysterious about the disappearance of the 180° term.

 

[speedlearner]

The textbook says as follows: An alternative method of describing the slope of a line with respect to the horizontal or x-axis utilizes the angle which the line makes with the axis. This angle of inclination is the counterclockwise angle whose initial side is the x-axis taken in the positive direction and whose terminal side lies on the line itself taken in the upward direction. Then is shows an illustration.

A few sentences later it says, "The slope of a line and the angle of inclination both give the direction of the line. The slope of a line is m = y2 -y1 / x2 - x1.

I've used the substitution method of mathematics for many years and I've done a comprehensive study of both Euclid and Descartes. I even studied Ptolomey's table of Chords. But none of these texts mention a reference angle. To this day I have no idea who invented the reference angle or why they invented such a mathematical term. I've asked many people and they are just as dumbfounded as I am.

Now in a regular mathematical equation, when you use the substitution method, you have to make sure both sides of the equation are equal. So when someone presents an example equation like this: 68 = 180 - 112, you could substitute the number 68 for 180 - 112. But in this equation I couldn't just remove the number 180. Otherwise the equation would look like this: 68 = -112. And as we all know 68 does not equal -112. Now can you see my confusion?

 

[Ray Vickson]

The textbook says as follows: An alternative method of describing the slope of a line with respect to the horizontal or x-axis utilizes the angle which the line makes with the axis. This angle of inclination is the counterclockwise angle whose initial side is the x-axis taken in the positive direction and whose terminal side lies on the line itself taken in the upward direction. Then is shows an illustration.

A few sentences later it says, "The slope of a line and the angle of inclination both give the direction of the line. The slope of a line is m = y2 -y1 / x2 - x1.

I've used the substitution method of mathematics for many years and I've done a comprehensive study of both Euclid and Descartes. I even studied Ptolomey's table of Chords. But none of these texts mention a reference angle. To this day I have no idea who invented the reference angle or why they invented such a mathematical term. I've asked many people and they are just as dumbfounded as I am.

Now in a regular mathematical equation, when you use the substitution method, you have to make sure both sides of the equation are equal. So when someone presents an example equation like this: 68 = 180 - 112, you could substitute the number 68 for 180 - 112. But in this equation I couldn't just remove the number 180. Otherwise the equation would look like this: 68 = -112. And as we all know 68 does not equal -112. Now can you see my confusion?

I am absolutely sure the book does not say
m = y_2 - \frac{y_1}{x_2} - x_1
as you wrote. Use parentheses!

 

[speedlearner]

I apologize for my mistake. Let's try that again. It actually says m = (y2 - y1)/(x2 - x1). Now will you answer my question?

 

[Ray Vickson]

I apologize for my mistake. Let's try that again. It actually says m = (y2 - y1)/(x2 - x1). Now will you answer my question?

I'm not sure I really understand what your question is. Could it be that you are confusing slope and angle? Slope = tan(angle) and angle = arctan(slope).

 

[speedlearner]

As I've stated previously, I've used the substitution method for many years. So when I say 68 = 180 - 112, I can also say 68 = 68 because I can substitute the number 68 in the place of the mathematical statement 180 - 112. So when Morris Kline says tan(A) = tan(180 - B), I can agree with that. After all, in his illustration, A = 180 - B. But I get stuck when he says tan(A) = tan(180 - B) = -tan(B). The reason I get stuck is the same reason I would get stuck saying 68 = 180 - 112 = -112. After all, I know that 68 will never equal -112. What I'm asking is what happens to the number 180.

 

[Mark44]

The textbook says as follows: An alternative method of describing the slope of a line with respect to the horizontal or x-axis utilizes the angle which the line makes with the axis. This angle of inclination is the counterclockwise angle whose initial side is the x-axis taken in the positive direction and whose terminal side lies on the line itself taken in the upward direction. Then is shows an illustration.

A few sentences later it says, "The slope of a line and the angle of inclination both give the direction of the line. The slope of a line is m = y2 -y1 / x2 - x1.

When you write this as text on a single line, you need parentheses, like so:
m = (y2 -y1) / (x2 - x1)

What you wrote would be interpreted as
m = y2 - $\frac{y1}{x2}$ - x1

I've used the substitution method of mathematics for many years and I've done a comprehensive study of both Euclid and Descartes. I even studied Ptolomey's table of Chords. But none of these texts mention a reference angle. To this day I have no idea who invented the reference angle or why they invented such a mathematical term. I've asked many people and they are just as dumbfounded as I am.

I'm not sure how useful it is to study "just" geometry, as presented in Euclid and Ptolemy. The combining of the geometry of the Greeks and the algebra of the Arabs and Indians) in analytic geometry was a large step forward in the understanding of mathematics of the time.

With that in mind, it's helpful to think in terms of the unit circle for the trig functions. For a given angle in its standard position, one ray is along the positive x-axis from (0, 0) to (1, 0), and the other ray extends to a point (x, y) on the unit circle. If the angle is acute (lies in Quadrant I), the reference angle is the same as the angle. For an angle such as 135°, for which the terminal ray is in Quadrant II, the terminal ray hits the unit circle at (-√2/2, √2/2). For this angle, the reference angle is the one made by the terminal ray and the negative x-axis, and has a measure of 45°.

All of the trig functions of 135° are the same in absolute value as for 45°, but fairly obviously, some of them are negative in value. For example, sin(135°) and sin(45°) are equal in value, but the cosines (and tangents) of the two angles are opposite in sign.

Now in a regular mathematical equation, when you use the substitution method, you have to make sure both sides of the equation are equal.

It's not really a substitution "method." When you substitute one expression for another, you are merely replacing something by something else that has the same value.

So when someone presents an example equation like this: 68 = 180 - 112, you could substitute the number 68 for 180 - 112. But in this equation I couldn't just remove the number 180. Otherwise the equation would look like this: 68 = -112. And as we all know 68 does not equal -112. Now can you see my confusion?

Yes. Kline is saying two things when A = 180° - B:
1. tan(A) = tan(180° - B). This part is clear to you, I believe. A and 180° - B are equal expressions, so they both have the same tangent value.
2. tan(A) = -tan(B). This is the part you're having trouble with. As I said in my first post, where this comes from is clear if you have the reference angles to draw on (which Euclid and Ptolemy didn't have).

Let's fall back to my example, with A = 45° and B its supplement (= 135°). Notice that these angles add to 180°, a so-called straight angle.

Angle A
Reference point on unit circle: ( √2/2, √2/2)

Angle B
Reference point on unit circle: ( -√2/2, √2/2)

On the unit circle, the x-coordinate gives the cosine of the angle, and the y-coordinate gives the sine of the angle. The ratio of the y-coordinate over the x-coordinate gives the tangent of the angle, so for this example, tan(A) = 1, and tan(B) = -1 = - tan(A).

It's easy enough to show using ordinary geometry, that this equation holds for arbitrary angles A and B that are supplementary.

 

[speedlearner]

I agree that the reference angle is the part that Mr. Kline left out. It isn't mentioned in Euclid, Ptolemy, or even Descartes work. As a matter of fact, no one I've spoken to so far has any idea as to the origin of reference angle. I'm sure no one just went to bed one night, woke up, and said I think reference angles are a great idea. Why Mr. Kline never mentioned the origin of reference angles is beyond me. It's even more amazing to me that so many people know nothing about the origin of reference angles. So where do they come from? And how do you substitute reference angles for equations like 180 - B?

 

[speedlearner]

I will also mention that no one that I've spoken to so far knows when negative numbers were introduced into the Cartesian Coordinate system. Was it Fermat? Was it Issac Newton? Maybe someone here knows the answer.

 

[Ray Vickson]

I will also mention that no one that I've spoken to so far knows when negative numbers were introduced into the Cartesian Coordinate system. Was it Fermat? Was it Issac Newton? Maybe someone here knows the answer.

Google 'negative number'. According to the Wikipedia entry, "Negative numbers appeared for the first time in history in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han Dynasty (202 BC – AD 220)". Later, in the section entitled History it says "European mathematicians, for the most part, resisted the concept of negative numbers until the 17th century, although Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of Liber Abaci, AD 1202) and later as losses (in Flos)." The article has more to say on the matter, but that should be enough to get you started.

Google is your friend.

 

[Mark44]

I agree that the reference angle is the part that Mr. Kline left out.

Since this is a problem from a calculus text, the author probably assumed some knowledge of trigonometry on the part of the reader.

It isn't mentioned in Euclid, Ptolemy, or even Descartes work.

I wouldn't expect any discussion of the unit circle or reference angles in works by the ancient Greeks, but Descartes might have used these concepts in his work of developing analytic geometry. As you might know, the Cartesian coordinate system is named after him.

As a matter of fact, no one I've spoken to so far has any idea as to the origin of reference angle.

Nor do I, but I have never lost any sleep over it.

I'm sure no one just went to bed one night, woke up, and said I think reference angles are a great idea. Why Mr. Kline never mentioned the origin of reference angles is beyond me.

Why? It would be impossible for the writer of a calculus text to include the origin of every term used in the book.

It's even more amazing to me that so many people know nothing about the origin of reference angles.

Again, why is this so amazing? For most people it's enough to know about them and how to use them, and of little or no importance to know when they came about or who thought of them.

So where do they come from? And how do you substitute reference angles for equations like 180 - B?

180 - B is not an equation - it's an expression. More precisely, it is 180° - B. In posts 2 and 8 I explained to the best of my ability how I believe Kline arrives at tan(A) = -tan(B) where A + B = 180°. Please reread those posts and see if they answer your question about his proof.