Length-Proportions that Depend on Rotations

UNH Mathematics Center

1  The proportionality measures

Trigonometry begins here: we have a length, or a vector, that's positioned at an angle. We want to describe the vertical and horizontal components of the angled vector. This is the sketch we began with:

Below are some diagrams of vectors placed at an angle:

the red vectors

are placed at an angle

the green vectors

are their vertical components

the blue vectors

are their horizontal components

The angular vectors are of different lengths, but in each sketch the angle with the horizontal is the same. So, the features that are the same in the three sketches should be those that depend on the angle at which the red vector is placed.

Each of the three trig functions sine, cosine, tangent records a feature of the sketches above that depends only on the angle of inclination q of the red (slanted) vector:

tanq

is the slope of the red vector itself:

(horizontal vector¢s length)(tanq) = (vertical vector¢s length)

sinq

is the multiplier that converts the slanted vector's length to that of its vertical component:

(slanted vector¢s length)(sinq) = (vertical vector¢s length)

cosq

is the multiplier that converts the slanted vector's length to that of its horizontal component:

(slanted vector¢s length)(cosq) = (horizontal vector¢s length)

1.1  Finding values for the proportionality measures sin(q), cos(q), tan(q)

The thing that worries most students is that it's not at all apparent how the values of the tangent, sine, and cosine functions are produced.

Even though there are tables of values for these functions, it's disconcerting not to have some arithmetical way of finding them. Although that is indeed the situation, we're not really left without any clues ...

2  Trig values for first-quadrant rotations

• For our own convenience we'll assume that the red, slanted vector's length is always 1.
• q is the angle of inclination of the slanted vector - its angle with the horizontal.

If the inclination angle were zero, the red vector would be horizontal - so that it would coincide with the blue vector. As the angle q begins to increase, the red vector begins to rotate counterclockwise ...

until, by the time q reaches p/2 (one right angle) the red vector will be vertical, and coincide with the green one.

This lets us see how the trig functions sine, cosine, tangent change as the angle q increases from 0 to p/2:

sinq increases from 0 to 1

Before the rotation began, the green vector was zero. It increases as the angle grows, until when q is a right angle the red vector and the green one are the same.

cosq decreases from 1 to 0

Before the rotation began, the red vector and the blue one were the same. The blue vector shrinks as the angle grows, until when q is a right angle the blue vector is zero.

tanq increases from 0 to ... ?? ...

(Shall we say, to ¥?) Before the rotation began, the red vector was the same as the blue horizontal one. Horizontal lines have zero slope. As the angle grows, the slope of the red vector increases indefinitely. When q is finally a right angle, the red vector is vertical, so its slope is undefined.

2.1  Landmark trig values in the first quadrant

There are two familiar triangles (you probably learned about them in high school geometry) that will start us off with some ratios of side-lengths of right triantles:

30°-60°-90° right triangles

have shorter sides that are half as long as their hypotenuses. If the hypotenuse is 1 unit long, the side opposite the 30° angle is 0.5 units long. The Pythagorean theorem tells us that the remaining side must be Ö3/2 units long.

45°-45°-90° right triangles

are isosceles. If its hypotenuse is 1 unit long, the Pythagorean theorem tells us that its perpendicular sides are each Ö2/2 units long.

Because of the ratios of side-lengths of right triangles, these two triangles are actually enough to give us excellent landmarks for the values of the trig functions in the first quadrant:

Rotation	Rotation	sine values	cosine values	tangent values
(radians)	(degrees)	increase:	decrease:	increase:
0	0°	0	1	0
p/6	30°	1/2 = 0.5	Ö3/2 @ 0.866	Ö3/3 @ 0.577
p/4	45°	Ö2/2 @ 0.707	Ö2/2 @ 0.707	1
p/3	60°	Ö3/2 @ 0.866	1/2 = 0.5	Ö3 @ 1.732
p/2	90°	1	0	not defined

Become familiar with these numbers! Knowing them will let you make good estimates for other angles.

Astound your friends:

Suppose we have a vector placed at an inclination of 55°. It is 2 units long. What is its vertical component?

We know:

• Its vertical component has length 2·sin(55°).
• The sine of 55° must be bigger than sin(45°) yet smaller than sin(60°) ... which means, between Ö2/2 @ 0.707 and Ö3/2 @ 0.866. Let's use 0.8 as a quick estimate.
• Thus, this vector's vertical component is roughly 2×0.8 = 1.6 units long.

That's a pretty good estimate. Your friend with a calculator has just arrived at 1.638, and is standing there looking impressed ...

Another example:

We have a 30-foot ladder. At what angle should we place it, so that it will reach a second-floor window 17 feet above the ground?

We know:

• We find the vertical distance by multiplying the diagonal length by sinq.
• What can we multiply 30 feet by, to get 17 feet? The multiplier we need is of course the fraction 17/30, which is the same as 1.7/3 @ 0.57. That's between sin(30°) = 0.5 and sin(45°) @ 0.707. It seems a little closer to sin(30°).
• Let's estimate 35°.

While you set the ladder at a 35° angle your friend with the calculator has figured that it will actually reach to 17.207 feet. Pretty close! (Don't lean the ladder against the glass.)

3  Bigger and Better Trig Functions

We went out of our way in the section on angles and rotations to define rotations so that they could be bigger than the angles we would find in triangles - or, for that matter, so that they could be negative.

We will want to define values of our functions sine, cosine, and tangent for these other rotations also. The plan will be:

• For angles of the right size to belong in triangles, the trig functions sine, cosine, tangent will have their same, accustomed values.
• These three trig functions will also be defined for rotations - of any size, and whether positive or negative. (Excepting, as before, rotations where the final side is vertical so that no tangent value can be defined.)

3.1  A first-quadrant example: q = p/6

This rotation, p/6, is one of our familiar first-quadrant angles. You will remember that we listed values for it in our table:

Rotation	Rotation	sine value	cosine value	tangent value
p/6	30°	1/2 = 0.5	Ö3/2 @ 0.866	Ö3/3 @ 0.577

In this sketch you can see that the green vertical vector points to sin(p/6) = 1/2 and the blue horizontal one points to cos(p/6) = Ö3/2 @ 0.866.

The slope of the red vector (the rotation's final side) is Ö3/3. The red vector is clearly not as steep as one whose slope is 1.

You should make a point of seeing these trig values as a feature of the sketch. Learning to do this will free you from endless, pointless, tedious memory tasks.

3.2  Another first-quadrant example: q = p/4

Before we venture beyond the familiar first quadrant, we'll illustrate the trig functions sine, cosine, tangent for q = p/4.

You can look up their values in the table:

Rotation	Rotation	sine value	cosine value	tangent value
p/4	45°	Ö2/2 @ 0.707	Ö2/2 @ 0.707	1

and, more importantly, you can see them in the picture.

The rotation's final side (the red vector) has length 1, so that its vertical and horizontal component vectors have length sin(p/4) and cos(p/4) respectively.

Notice that

• the vertical component (the green vector) points directly to sin(p/4) @ 0.707, while
• the horizontal component (the blue vector) points directly to cos(p/4) @ 0.707.
• The slope of the red vector is 1. That's how we can see (and not memorize!) that tan(p/4) = 1.

3.3  A second-quadrant example: q = 2p/3

Now we are ready to make the Big Leap, and extend the functions sine, cosine, tangent so that they apply to rotations beyond the first quadrant.

We'll use 2p/3 as a first example. It's a rotation in the second quadrant, and we can think of it as having begun at the horizontal ''3 o'clock'' position on the circle, and then rotating 2p/3 radians in the positive (counterclockwise) direction, so that it ends as the red vector in the sketch to the left. The red vector is the rotation's ''final side.''

The red vector's vertical and horizontal components are the green and blue vectors, respectively. We will define the sine, cosine and tangent values of this rotation in terms of the red, green and blue vectors so that they match the interpretations we made for first-quadrant rotations.

cos(2p/3)

is the multiplier that converts the length of the slanted vector to that of its horizontal component.

Now you can see how convenient it was to keep the length of the slanted (red) vector fixed at 1! You can see that the blue vector's length is 0.5, and that it's pointed in the negative direction. The blue vector points right to the number cos(2p/3) = -1/2.

sin(2p/3)

is the multiplier that converts the length of the slanted vector to that of its vertical component.

The green vector - the vertical component of the slanted (red) vector - has length sin(2p/3). The green vector points to sin(2p/3) = Ö3/2 @ 0.866.

tan(2p/3)

is the slope of the red vector! It's also the quotient - the sine, divided by the cosine - so that tan(2p/3) = -Ö3.

You can see that it looks right. Slopes of ''downhill'' lines are negative. Any rotation whose final side is in the second quadrant will have a ''downhill'' final side, and hence a negative tangent. You no longer have to memorize that tangent values are negative in the second quadrant!

3.3.1  Family resemblances among cousins: 2p/3 and p/3

Our second-quadrant rotation 2p/3 has a cousin in the first quadrant, which you will recognize. It's the mirror image of the first-quadrant rotation p/3:

We've added the cousin-rotation p/3 to the sketch: its final side is pink, and its horizontal component is light blue. You will notice some things right away:

• 2p/3 and p/3 share the same vertical component. So, they have the same sine value: sin(2p/3) = sin(p/3) = Ö3/2 @ 0.866.
• The blue, and light blue, horizontal components have the same length but are pointed in opposite directions: cos(2p/3) = -0.5 while cos(p/3) = 0.5.
• The slopes of the red 2p/3 vector, and the pink p/3 vector, are negatives of each other: one is Ö3 and the other is -Ö3. The one in the first quadrant is positive, and the one in the second quadrant is negative, of course!

You can see from this sketch that because we know values of the trig functions sine, cosine, tangent at the ''familiar'' angles in the first quadrant, we can also find trig values for their cousins in the second quadrant. It's not even a matter of memorizing them. You just picture the second-quadrant rotation and think of its first-quadrant cousin ...

3.3.2  Three cousins: -5p/6, 5p/6 and p/6

• -5p/6, a rotation in the negative (clockwise) direction that starts at the usual position on the positive horizontal axis and ends with the red vector in the third quadrant;
• 5p/6, a rotation in the positive (counterclockwise) direction that ends with the pink vector in the second quadrant; and
• p/6, a familiar ''special rotation'' in the first quadrant

You will see from the symmetry that

• The two rotations: the red one, -5p/6 and its cousin, the pink 5p/6 in the third quadrant, have the same horizontal component. The blue vector points to the cosine, -Ö3/2, that both 5p/6 and -5p/6 share.
• The light-blue vector on the horizontal axis points right to cos(p/6) = Ö3/2, the negative of the cos(5p/6).
• The sine values are opposite in the second and third quadrants: sin(-5p/6) = -1/2, while sin(5p/6) = 1/2. However, 5p/6 and p/6 share the same sine value: sin(5p/6) = sin(p/6) = 1/2.
• The tangent values are slopes of the rotations' final sides. tan(-5p/6) = tan(p/6) = -Ö3/3. On the other hand, tan(5p/6) = Ö3/3 = -tan(-5p/6).

3.4  Periodicity of the trig functions

Another thing we see from these pictures is that the value of a trig function at a particular rotation depends only on the rotation's final side.

That is, although two rotations can be different and yet have the same final side, their trig values will be the same. This isn't shocking. It's just that the trig functions are periodic.

For example: p/3, -5p/3 and 7p/3 are three different rotations. Certainly when we think of them as numbers, they are different! But the values of the trig functions are the same:

sin(p/3) = sin(-5p/3) = sin(7p/3) = Ö3/2

cos(p/3) = cos(-5p/3) = cos(7p/3) = 1/2

tan(p/3) = tan(-5p/3) = tan(7p/3) = Ö3

4  Finally! the Other Trig Functions, and the ''minus-one trap''

No, we haven't forgotten. There are three trig functions we haven't named yet: cosecant, secant, cotangent. They have always been there in the background, as their values are the reciprocals of the functions sine, cosine, tangent (in that order).

There is a notational matter that you won't want to be confused about, based on the exponent -1 which is such a convenient notation that (unfortunately) it's used in more than one way. It is too bad that things are the way they are. But that's the way they are. If you're aware of the two different uses of the exponent -1, you won't be confused by them:

• Sometimes the number -1 is meant as a real exponent, and introduces a reciprocal fraction. For instance, 2^-1 = 1/2.
• At other times the ''exponent'' -1 is used in functional notation, so that the context is that of an inverse function. For instance, if the function f has an inverse we would give the inverse function the name f^-1. In this situation, if
  

  f(2) = 3

  then
  

  f-1(3) = 2

  In other words, if (2,3) is an ordered pair in the function f, then (3,2) is an ordered pair belonging to the inverse function, f^-1.

You won't want to confuse the reciprocal of the sine function with the inverse-sine function. Both are good, useful functions!

• If we write (sin(p/3))^-1, we mean the reciprocal of the number sin(p/3). That would be 2/Ö3, which is csc(p/3). (The abbreviation of ''cosecant'' is csc.)

• However, if the -1 is used to indicate an inverse function, as in
  

  sin-1(Ö3/2) = p/3

  we are not talking about reciprocal numbers at all. Rather, we are saying that ''the'' number whose sine is Ö3/2, is p/3.

  The functional notation ''sin^-1'' is alternatively written ''arcsine,'' and is pronounced either arcsine or inverse-sine. You will probably find it on your calculator as the ''second function'' associated with the sine key. This function key returns one particular number: the number between -p/2 and p/2, whose sine is the number you entered.

  Remember, ''arcsine'' is a function. It cannot return more than one value! It will return its value in either radians or degrees, depending on how you've set your calculator.

  Try it! Use the key ''sin^-1 with Ö3/2 and see what you get. The calculator will return either 60 (if it's in degrees) or a decimal approximation 1.0472... for p/3. (Remember that p is a number, which your calculator will display as 3.14159....)

And finally there is the cotangent function, whose values are reciprocals of the tangent function. And once again: the notation tan^-1 will mean the arctangent function, which is different from the cotangent function. A number's arctangent value is the rotation between -p/2 and p/2, whose tangent is the number in question.