Angles and Rotations

Measuring rotations
In previous chapters we explored the meaning of expressions such as sin 30°, cos 45° and tan 60°. In this chapter and the next we show how we can use expressions such as sin 180°, tan 300° or even sin 1000°.
But what might 1000° measure? Certainly it is not the measure of the angle of a triangle. These can only be between oo and 180° (acute, right or obtuse). Nor can it be the measure of an angle (or an arc) in a circle.

These can only be between oo and 360°.
If you have ever owned toy electric trains, you may have set up the tracks in a circle, and run the trains around the circle. The diagram below shows a circular track. If a train starts at point A, travels around the circle, and arrives back at point A, we say that it has made one full rotation around the circle.

Since we divide a circle into 360 degrees, it is natural to say that the train has rotated around the circle by 360 degrees.
Now suppose the train continues past point A, and travels around the circle again. Then we can say that it has rotated through more than 360 degrees.

If it travels around the circle twice, returning to point A, we say that it has rotated 360 + 360 = 720 degrees.

And if it travels a bit further around the circle, along an arc measuring 280°, we say that it has rotated 720° + 280° = 1000°:

Here is another example. Look at the hour hand of a clock. In 12 hours it has made a full rotation, or rotated by 360°.

But this time the rotation is clockwise (by definition!), while our train was rotating counterclockwise. In a plane, there are two different directions of rotation, and it turns out to be important to distinguish between them.
Mathematicians call a counterclockwise rotation positive and a clockwise rotation negative. So we say that in 12 hours, the hour hand of a clock performs a rotation of -360°.

Rotation and angles.

Picture a circle of radius 1, with its center at the origin of a system of coordinates:

We take an acute angle with one leg along the x-axis. The other leg will end up someplace in the first quadrant. If the measure of this acute angle is, say, 40°, then we can get from point Q to point P by rotating through 40°.

So we can associate angles with rotations. Even if the rotation exceeds 180″, we sometimes talk about the “angle” instead of the “rotation.” The figure below gives some examples.

Trigonometric functions for all angles
Let us look again at what we mean by sin 40°. We will do so in such a way that it will help us understand what is meant by sin 300°, cos 1100°, or tan(-240°).
We draw a circle of radius r centered at the origin of coordinates. To

Trigonometric function for all angles.

Similarly, we can write

So far we have said nothing new.
Or have we?

We can use this observation to extend our definitions of sine and cosine to our new angles, which measure rotations. Suppose a point P starts at position (r, 0), and rotates through an angle ot.

If P has coordinates (x, y), we define cos a and sin a by writing

Note that these new definitions give the same values as the old definitions when ot is an acute angle.
Example 29 Find the numerical values of sin 130° and cos 130°.
Note that by Chapter 3, page 71, we already know that sin 130° = sin (180° – 130°) = sin 50°, so in fact, this quantity had been defined already.
But let us see if our new definition gives the same result.

The circle in the diagram has radius r, and the point P has rotated through an angle of 130° from point Q. If the coordinates of P are (x, y), our new definition tells us that

Now we look at right triangle 0 P R, in which !. P 0 R = 50°, and note that

So this is the value of sin 130°.
Triangle 0 P R will also give us the numerical value of cos 130°, but we must be careful. Since the x-coordinate of point P is negative, we must write

Thus, cos 130° = -cos (180° – 130°) = -cos 50°. This value agrees with the one given by the definition on page 79.

It is important to note that the result above does not depend on the length of 0 P. We can choose a circle of any radius and draw the corresponding diagram for a 130° angle. Triangle 0 P R will always have the same angles, and the computation will be the same.

Example 30 What are the values of cos 210° and sin 210°?
Solution. Since the radius of the circle will not matter, we are free to choose, for example, a circle of radius 1. Then our new definitions lead to the diagram below:

The geometry of a 30-60-90 triangle shows that the coordinates of point P

Example 31 Find the values of cos 360° and sin 360°.
Solution. We choose a circle of radius 1. For a rotation of 360°, the coordinates of point P are (1, 0). Therefore, cos 360° = 1 and sin 360° =Q.

Now that we have definitions for sine and cosine of any angle, we can make definitions for the other trigonometric functions of these angles.
For any angle a,

Example 32 Find the numerical value of tan 21 oo. Solution.

From the results of Example 30, we have that

Note that this value is positive.

Our new definitions of sine and cosine give values for any angle a. But this is not quite true for our new definitions of tangent, cotangent, secant and cosecant, because they involve division. We must be sure that we are not dividing by 0.
Indeed, we will not define tan a if cos a = 0. Expressions such as tan 90°, tan 270°, and tan ( -90°) must remain undefined.
For similar reasons, we cannot define cot oo, or esc 180°.

Calculations with angles of rotations.

Let us look back at our original picture of an angle in a circle:

Originally, we thought of this as an angle of 40°. But a diagram of a 400° angle would look exactly the same, as would a diagram for 760° or -320°.

The diagram will look the same for any two angles which differ by a full rotation. Therefore, sin a = sin (a + 360°).
Similarly, cos a= cos (a+ 360°) for any angle a. These observations allow us to find the sine, cosine, tangent, or cotangent of very large angles easily.

Example 33 What is cos 1140°?
Solution. If we divide 1140 by 360, the quotient is 3 and the remainder
is 60, that is, 1140 = 3 · 360 + 60. So cos 1140° = cos (3 · 360 + 60°) = cos60o = 1/2.

Example 34 Is the sine of 100,000° positive or negative?
Solution. If we divide 100,000 by 360, we get 277, with a remainder of 280. So the sine of 100,000° is the same as sin 280°. Since the position of point P is in the fourth quadrant, its y-coordinate is a negative number.
The sine of 100,000° is therefore negative.  You can check the logic of these solutions using your calculator, which
already “knows” if the sine of an angle is positive or negative.

That is, the people who designed it did exercises like yours before they built the calculator.
But it is also important to be able to “predict” certain values of the trigonometric functions, or at least tell whether their values will be positive or negative. It’s not difficult to see that if point P ends up in quadrant I, all functions of the angles are positive.

If point P lies in quadrant II, the sine and cosecant are positive, and all other functions are negative, and so on.

Example 35 Find sin 300°.
Solution. Point P, having rotated through 300°, will end up in quadrant IV. So sin 300° is negative. Furthermore, the angle made by one side and the x-axis is 60°.

b) If sin a = -5/13, in what quadrant can a lie? What are the possible values of cos a?

sin (-a)=- sin a for any angle a.

Similarly, we find the following:
tan (-a) – -tan a
cot (-a) = – cot a .
However, the cosine function is different. We have

cos (-a) = cos a .
In general, we can distinguish two type of functions.

A function is called even if, for every x, f (-x) = f (x ).
A function is called odd if,for every x, f(-x) = -f(x).
So, for example, the functions

are all even, while the functions

are all odd. The following functions are neither even nor odd:

cos x is an even function, while sinx, tanx, and cos x are odd functions.
This may be the reason that some mathematicians prefer to work with the cosine function, rather than the sine.