12 ## ANGLES AND THEIR MEASUREMENTTRIGONOMETRY, 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. Angles 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 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 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. Degree measure 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 Standard position 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 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?
(Skill in Arithmetic, Equivalent Fractions, Question 5.) 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°? 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 Why do we name the quadrants in the counter clockwise direction? Because in what we call the "first" quadrant, the algebraic signs of 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°
Coterminal angles 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.
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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