ANGLES AND THEIR MEASUREMENT
TRIGONOMETRY, as it is actually used in calculus and science, is not about solving triangles. It becomes the mathematical description of things that rotate or vibrate, such as light, sound, the paths of planets about the sun, or satellites about the earth. It is necessary therefore to have angles of any size, and to extend to them the meanings of the trigonometric functions. We do that in Topic 15.
An angle is the opening that two straight lines form when they meet.
When the straight line FA meets the straight line EA, they form the angle we name as angle FAE. Letter A, which we place in the middle, labels the point where the two lines meet, and is called the vertex of the angle. When there is no confusion as to which point is the vertex, we may speak of "the angle at the point A," or simply "angle A."
The two straight lines that form an angle are called its sides. And the size of the angle does not depend on the lengths of its sides. We can see that in the figure above. For if the point C is in the same straight line as FA, and B is in the same straight line as EA, then angles CAB and FAE are the same angle.
Now, to measure an angle, we place the vertex at the center of a
circle (we call that a central angle), and we measure the length of the arc -- that portion of the circumference -- that the sides intercept. We then determine what relationship that arc has to the entire circumference, which is an agreed-upon number. (In degree measure that number is 360; in radian measure it is 2π.)
The measure of angle A, then, will be length of the arc BC relative to the circumference BCD -- or the length of arc EF relative to the circumference EFG. For in any circles, equal central angles determine a unique ratio of arc to circumference. (See the theorem of Topic 14. It is stated there in terms of the ratio of arc to radius, but the circumference is proportional to the radius: C = 2πr.)
There are two systems for measuring angles. One is the well-known system of degree measure. The other is the strictly mathematical system called radian measure, which we take up in the next Topic.
To measure an angle in degrees, we imagine the circumference of a circle divided into 360 equal parts, and we call each of those equal parts a "degree." Its symbol is a small 0: 1° -- "1 degree." The full circle, then, will be 360°. But why the number 360? What is so special about it? Why not 100° or 1000°?
The answer is two-fold. First, 360 has many divisors, and therefore it will have many whole number parts. It has an exact half and an exact third -- which a power of 10 does not have. 360 has a fourth part, a fifth, a sixth, and so on. Those are natural divisions of the circle, and it is very convenient for their measures to be whole numbers. (Even the ancients didn't like fractions)
Secondly, 360 is close to the number of days in the astronomical year: 365.
The measure of an angle, then, will be as many degrees as its sides include. To say that angle BAC is 30° means that its sides enclose 30 of those equal divisions. Arc BC is of the entire circumference.
So, when 360° is the measure of a full circle, then 180° will be half a circle. 90° -- one right angle -- will be a quarter of a circle; and 270° will be three quarters of a circle: three right angles.
Let us now see how we deal with angles in the x-y plane.
We say that an angle is in standard position when its vertex A is at the origin of the coördinate system, and its Initial side AB lies along the positive x-axis. We say that AB has "swept out" the angle BAC, and that AC is its Terminal side.
We now think of the terminal side AC as rotating about the fixed point A. When it rotates in a counter-clockwise direction, we say that the angle is positive. But when it rotates in a clockwise direction, as AC', the angle is negative.
When the terminal side AC has rotated 360°, it has completed one full revolution.
Problem 1. How many degrees corresponds to each of the following?
To see the answer, pass your mouse over the colored area.
a) A third of a revolution A third of 360° = 360° ÷ 3 = 120°
b) A sixth of a revolution 360° ÷ 6 = 60°
c) Five sixths of a revolution 5 × 60° = 300°
d) Two revolutions 2 × 360° = 720°
e) Three revolutions 3 × 360° = 1080°
f) One and a half revolutions 360° + 180° = 540°
Example 1. 30° is what fraction of a circle, or of one revolution?
Problem 2. What fraction of a revolution is each of the following?
Example 2. If the diameter of a circle is 16 cm, how long is the arc intercepted by a central angle of 45°?
Answer. 45° is one eighth of a full circle. (It is half of 90 °, which is one quarter.) Now, the full circumference of this circle is
C = πD = 3.14 × 16 cm.
(Topic 9.) The intercepted arc is one eighth of the circumference:
3.14 × 16 ÷ 8 = 3.14 × 2 = 6.28 cm
Problem 3. If the diameter of a circle is 20 in, how long is the arc intercepted by a central angle of 72°?
We saw in Problem 2c) that 72° is one fifth of a circle. The circumference of this circle is C = πD = 3.14 × 20 in. The intercepted arc is one fifth of this: 3.14 × 20 ÷ 5 = 3.14 × 4 = 12.56 in.
The four quadrants
The x-y plane is divided into four quadrants. The angle begins in its standard position in the first quadrant ( I ). As the angle continues -- in the counter-clockwise direction -- we name each succeeding quadrant.
Why do we name the quadrants in the counter clockwise direction? Because in what we call the "first" quadrant, the algebraic signs of x and y are positive.
Problem 4. In which quadrant does each angle terminate?
a) 15° I b) −15° IV c) 135° II
d) 390° I. 390° = 360° + 30°
e) 100° II
f) −460° III. −460° = −360° − 100°
Angles are coterminal if, when in the standard position, they have the same terminal side.
For example, 30° is coterminal with 360° + 30° = 390°. They have the same terminal side. That is, their terminal sides are indistinguishable.
Any angle θ is coterminal with θ + 360° -- because we are just going around the circle one complete time.
−90° is coterminal with 270°. Again, they have the same terminal side.
Notice: 90° plus 270° = 360°. The sum of the absolute values of those coterminal angles completes the circle.
Problem 5. Name the non-negative angle that is coterminal with each of these, and is less than 360°.
a) 360° 0°
b) 450° 90°. 450° = 360° + 90°
c) −20° 340°
d) −180° +180°
e) −270° 90°
f) 720° 0°. 720° = 2 × 360°
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