As you see, the word itself refers to three angles – a reference to triangles. Trigonometry is primarily a branch of mathematics that deals with triangles, mostly right triangles. In particular the ratios and relationships between the triangle’s sides and angles. It has two main ways of being used: 1. In geometryIn its geometry application, it is mainly used to solve triangles, usually right triangles. That is, given some angles and side lengths, we can find some or all the others.
For example, in the figure below, knowing the height of the tree and the angle made when we look up at its top, we can calculate how far away it is (CB). (Using our full toolbox, we can actually calculate all three sides and all three angles of the right triangle ABC). 2. AnalyticallyIn a more advanced use, the trigonometric ratios such as as Sine and Tangent, are used as functions in equations and are manipulated using algebra. In this way, it has many engineering applications such as electronic circuits and mechanical engineering.
In this analytical application, it deals with angles drawn on a coordinate plane, and can be used to analyze things like motion and waves. Chapter-1Angles in the Quadrants( Some basic Concepts)In trigonometry, an angle is drawn in what is called the “standard position”. The vertex of the angle is on the origin, and one side of the angle is fixed and drawn along the positive x-axis. Names of the partsThe side that is fixed along the positive x axis (BC) is called the initial side. To make the angle, imagine of a copy of this side being rotated about the origin to create the second side, called the terminal side.
The amount we rotate it is called the measure of the angle and is measured in degrees or radians. This measure can be written in a short form: mABC = 54° which is spoken as “the measure of angle ABC is 54 degrees”. If it is not ambiguous, we may use just a single letter to denote an angle. In the figure above, we could refer to the angle as ABC or just angle B. In trigonometry, you will often see Greek letters used to name angles. For example the letter ? (theta), but on this site we always use ordinary letters like A,B,C.
The measure can be positive or negativeBy convention, angles that go counterclockwise from the initial side are positive and those that go clockwise are negative. In the figure above, click on ‘reset’. The angle shown goes counterclockwise and so is positive. Drag A down across the x-axis and see that angles going clockwise from the initial side are negative. See Trig functions of large and negative angles The measure can exceed 360°In the figure above click ‘reset’ and drag the point A around counterclockwise. Once you have made a full circle (360°) keep going and you will see that the angle is greater than 360°.
In fact you can go around as many times as you like. The same thing happens when you go clockwise. The negative angle just keeps on increasing. Coterminal anglesIf you have one angle of say 30°, another of 390°, the two terminal sides will be in the same place (390 = 360+30). These two angles would then be called coterminal angles. They would be in the same place on the plane but have different measures (30° and 390°). Degrees and radiansThe measure of an angles can be expressed in degrees or radians, but in trigonometry radians are the most common. See Radians and Degrees. Recall than there are 2? adians in a full circle of 360°, so 1 radian is approximately 57°. In the figure above, click on “radians” to change units. | Standard position of an angle In trigonometry an angle is usually drawn in what is called the “standard position” as shown below. In this position, the vertex of the angle (B) is on the origin of the x and y axis. One side of the angle is always fixed along the positive x-axis – that is, going to the right along the axis in the 3 o’clock direction (line BC). This is called the initial side of the angle. The other side of the angle is called the terminal side.
Initial side of an angle In trigonometry an angle is usually drawn in what is called the “standard position” as shown below. In this position, the vertex of the angle (B) is on the origin of the x and y axis. One side of the angle is always fixed along the positive x-axis – that is, going to the right along the axis in the 3 o’clock direction (line BC). This is called the initial side of the angle. The other side of the angle is called the terminal side. Terminal side of an angle In trigonometry an angle is usually drawn in what is called the “standard position” as shown on the right.
In this position, the vertex (B) of the angle is on the origin, with a fixed side lying at 3 o’clock along the positive x axis. The other side, called the ‘terminal side’ is the one that can be anywhere and defines the angle. In the figure below, drag point A and see how the position of the terminal side BA defines the angle. Quadrantal Angle Definition: Angles in the standard position where the terminal side lies on the x or y axis. For example: 90°, 180° etc. A quadrantal angle is one that is in the standard position and has a measure that is a multiple of 90° (or ? /2 radians).
A quadrantal angle will have its terminal lying along an x or y axis. | Coterminal angles From co -“together”, terminal -“end position” Definition: Two angles are coterminal if they are drawn in the standard position and both have their terminal sides in the same location| Recall that when an angle is drawn in the standard position as above, only the terminal sides (BA, BD) varies, since the initial side (BC) remains fixed along the positive x-axis. If two angles are drawn, they are coterminal if both their terminal sides are in the same place – that is, they lie on top of each other.
In the figure above, drag A or D until this happens. If the angles are the same, say both 60°, they are obviously coterminal. But the angles can have different measures and still be coterminal. In the figure above, rotate A around counterclockwise past 360° until it lies on top of DB. One angle (DBC) has a measure of 72°, and the other (ABC) has a measure of 432°, but they are coterminal because their terminal sides are in the same position. If you drag AB around twice you find another coterminal angle and so on. There are an infinite number of times you can do this on either angle.
Either or both angles can be negative In the figure above, drag D around the origin counterclockwise so the angle is greater than 360°. Now drag point A around in the opposite direction creating a negative angle. Keep going until angle DBC is coterminal with ABC. You can see that a negative angle can be coterminal with a positive one. How to tell if two angles are coterminal. You can sketch the angles and often tell just form looking at them if they are coterminal. Otherwise, for each angle do the following: * If the angle is positive, keep subtracting 360 from it until the result is between 0 and +360. In radians, 360° = 2? radians) * If the angle is negative, keep adding 360 until the result is between 0 and +360. If the result is the same for both angles, they are coterminal. Why is this important? In trigonometry we use the functions of angles like sin, cos and tan. It turns out that angles that are coterminal have the same value for these functions. For example, 30°, 390° and -330° are coterminal, and so sin30°, sin390° and sin(-330°) and all have the same value (0. 5). Reference Angle: The smallest angle that the terminal side of a given angle makes with the x-axis is called reference angle.
Chapter-2 Measurement Of Angles 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. 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 2?. ) 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 = 2? r. ) There are two systems for measuring angles. One is the well-known system of degree measure. . 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 | 30 60| 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. Standard position We say that an angle is in standard position when its vertex A is at the origin of the coordinate 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. To cover the answer again, click “Refresh” (“Reload”). 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 ? 60° = 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? Answer. 30° is | 30 360| of a revolution:| 30 360| = | 3 36| = | 1 12| Problem 2. What fraction of a revolution is each of the following? a) 60° | 60 360| = | 6 36| = | 1 6| b) 45° | 45 360| = | 5 40| = | 1 8| c) 72° | 72 360| = | 8 40| = | 1 5| 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 = ? D = 3. 14 ? 16 cm. 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 = ? D = 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° III f) ? 460° III. ?460° = ? 360° ? 100° g) 710° | IV. 10° is 10° less than two revolutions, which are 720°. | 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 ? is coterminal with ? + 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° g) ? 200° | 160°| The Radian Measure THE RADIAN SYSTEM of angular measurement, the measure of one revolution is 2?. (In the next Topic, Arc Length, we will see the actual definition of radian measure. ) Half a circle, then, is ?. And, most important, each right angle is half of ? : | ? 2| . | Three right angles will be 3· | ? | = | 3? 2 | . | Five right angles will be | 5? 2 | . And so on. | Radians into degrees The following radian measures come up frequently, and the student should know their degree equivalents: ? 4| is half of | ? 2| , a right angle, and so it is equal to 45°. | Equivalently, | ? 4| is of one quarter of ? , or half of half of 180°. | ? 3| is a third of ? , and so is equal to 180° ? 3 = 60°. | ? 6| is a sixth of ? , and so is equal to 180° ? 6 = 30°. | 5? 4 | = 5· | ? 4 | = 5· 45° = 225°. | 2? 3 | is a third of 2?. A third of a revolution = 360° ? 3 = 120°. | Problem 1 .
Convert each of these radian measures into degrees. Problem 1. The student should know these. To see the answer, pass your mouse over the colored area. To cover the answer again, click “Refresh” (“Reload”). a) ? 180° | b) | ? 2| 90° | c) | ? 3| 60° | d) | ? 6| 30° | e) | ? 4| 45° | Problem 2. Convert each of these radian measures into degrees. a) | ? 8| | 22? °. | ? 8| is half of | ? 4| . | b) | 2? 5| | 72°. | 2? 5 | is a fifth of 2? | , which is a fifth of a 360°. | c) | 7? 4| | = 7· | ? 4 | = 7· 45° = 315°| d) | 9? 2| | = 9· | ? | = 9· 90° = 810°| e) | 4? 3| | = 4· | ? 3 | = 4· 60° = 240°| f) | 5? 6| | = 5· | ? 6 | = 5· 30° = 150°| g) | 7? 9| | | Problem 3. Evaluate the following. a) cos | ? 6| = | 2| | b) sin | ? 6| = | 1 2| | c) tan | ? 4| = | 1| | d) cot | ? 3| = | 1 | | e) csc | ? 6| = | 2 | | f) sec | ? 4| = | | Problem 4. In terms of radians, what angle is the complement of an angle ? ? | ? 2| ? | ? | Problem 5. A function of any angle is equal to the cofunction of its complement. Therefore, in terms of cofunctions: a) sin ? = | cos (| ? 2| ? | ? | )| | b) cot ? | tan (| ? 2| ? | ? | )| c) sec (| ? 2| ? ?)| = csc ? | Degrees into radians 360° = 2?. When we write 2? , we mean 2? radians, which is approximately 6. 28 radians. However, we normally omit the word radians. As we will see in the next Topic, Arc length, the radian measure can be any real number. Problem 6. The student should begin by knowing these. 0° = | 0 radians. | | 360° = | 2?. | | 180° = | ?. | | 90° = | ? 2| . | 45° = | ? 4| . | | 60° = | ? 3| . | | 30° = | ? 6| . | Example 1. Convert 120° into radians. Solution. We can go from what we know to what we do not know.
In the most important cases we can recognize the number of degrees as a multiple of 90°, or 45°, or 60°, or 30°; or as a part of 360°. Since 60° = | ? 3| , then| 120° = 2· 60° = 2· | ? 3| = | 2? 3| . | Or, since 120° is a third of 360°, which is 2? , then 120° = | 2? 3| . | Example 2. 225° = 180° + 45° = ? + | ? 4| = | 5? 4| . | Or, 225° = 5· 45° = 5· | ? 4| = | 5? 4| . | Problem 7. Convert each of the following into radians. a) 270° = | 3· 90° = | | 3? 2| | b) 210° = | 7· 30° = 7· | ? 6| = | 7? 6| c) 300° = | 5· 60° = 5· | ? 3| = | 5? 3| | d) 135° = | 3· 45° = 3· | ? | = | 3? 4| e) 720° = | 2· 360° = 2· 2? = 4? | f) 450° = | 5· 90° = 5· | ? 2| = | 5? 2| g) 36° = | A tenth of 360° = | 2? 10| = | ? 5| h) 72° = | 2· 36° = | 2? 5| 72° is thus a fifth of a revolution. i) 40° = | A ninth of 360° = | 2? 9| j) 80° = | 2· 40° = | 4? 9| As a last resort, proportionally, so that Example 3. Change 140° to radians. Solution. | 140 180| · ? | = | 7 9| · ? | = | 7? 9| ,| upon dividing both the numerator and denominator first by 10 and then by 2Coterminal angles Angles are coterminal if they have the same terminal side. ? is coterminal with ??.
They have the same terminal side. Notice that ? + ? = 2? , so that ? = 2? ? ? . . . . . . . . (1) Example 4. Name in radians the non-negative angle that is coterminal with ? | 2? 5| , and is less than 2?. | Answer. Let us call that angle ?. Then according to line (1), ? = 2? ? | 2? 5| = | 10? ? 2? 5| = | 8? 5| Problem 8. Name in radians the non-negative angle that is coterminal with each of the following, and is less than 2?. a) ? | ? 6| . | ? = 2? ? | ? 6| = | 12? ? ? 6| = | 11? 6| b) ? | 3? 4| . | ? = 2? ? | 3? 4| = | 8? ? 3? 4| = | 5? 4| c) ? | 4? 3| . | ? = 2? | 4? 3| = | 6? ? 4? 3| = | 2? 3| The multiples of ? Starting at 0, let us go around the circle a half-circle at a time. We will then have the following sequence, which are the multiples of ? : 0, ? , 2? , 3? , 4? , 5? , etc. The point to see is that the odd multiples of ? , ?, 3? , 5? , 7? , etc. are coterminal with ?. While the even multiples of ? , 2? , 4? , 6? , etc. are coterminal with 0. If we go around in the negative direction, we can make a similar observation. Problem 9. Name in radians the non-negative angle that is coterminal with each of the following, and is less than 2?. ) -? ? b) -2? 0 c) -3? ? d) -4? 0 e) -5? ? f) 3? ? g) 4? 0 h) 5? ? i) 6? 0 j) 7? ? IT IS CONVENTIONAL to let the letter s symbolize the length of an arc, which is called arc length. We say in geometry that an arc “subtends” an angle ? ; literally, “stretches under. ” Now the circumference of a circle is an arc length. And the ratio of the circumference to the diameter is the basis of radian measure. That ratio is the definition of ?. ?| = | C D| . | Since D = 2r, then ?| =| C 2r| or, C r| = | 2? | . | That ratio — 2? – of the circumference of a circle to the radius, is called the radian measure of 1 revolution, which are four right angles at the center. The circumference subtends those four right angles. Radian measure of ? = | s r| Thus the radian measure is based on ratios — numbers — that are actually found in the circle. The radian measure is a real number that names the ratio of a curved line to a straight, of an arc to the radius. For, the ratio of s to r does determine a unique central angle ?. | Theorem. | | In any circles the same ratio of arc length to radius| | | determines a unique central angle that the arcs subtend. Proportionally, if and only if ?1 = ? 2. We will prove this theorem below. Example 1. If s is 4 cm, and r is 5 cm, then the number | 4 5| , i. e. | s r| , is the| radian measure of the central angle. At that central angle, the arc is four fifths of the radius. Example 2. An angle of . 75 radians means that the arc is three fourths of the radius. s = . 75r Example 3. In a circle whose radius is 10 cm, a central angle ? intercepts an arc of 8 cm. a) What is the radian measure of that angle? Answer. According to the definition: ? = | s r| = | 8 10| = . 8| b) At that same central angle ? what is the arc length if the radius is b) 5 cm? Answer. For a given central angle, the ratio of arc to radius is the same. 5 is half of 10. Therefore the arc length will be half of 8: 4cm. Example 4. a) At a central angle of 2. 35 radians, what ratio has the arc to the radius? Answer. That number is the ratio. The arc is 2. 35 times the radius. b) In which quadrant of the circle does 2. 35 radians fall? Answer. Since ? = 3. 14, then | ? 2| is half of that: 1. 57. | 3? 2| = 3. 14 + 1. 57| = 4. 71. An angle of 2. 35 radians, then, is greater than 1. 57 but less that 3. 14. It falls in the second quadrant. = r? c) If the radius is 10 cm, and the central angle is 2. 35 radians, then how c) long is the arc? Answer. We let the definition of ? , ? = | s r| become a formula for finding s : s = r? | Therefore, s = 10 ? 2. 35 = 23. 5 cm Because of the simplicity of that formula, radian measure is used exclusively in theoretical mathematics. The unit circle Since in any circle the same ratio of arc to radius determines a unique central angle, then for theoretical work we often use the unit circle, which is a circle of radius 1: r = 1. In the unit circle, the length of the arc s is equal to the radian measure.
The length of that arc is a real number x. s = r? = 1· x = x. We can identify radian measure, then, as the length x of an arc of the unit circle. And it is here that the term trigonometric “function” has its full meaning. For, corresponding to each real number x — each radian measure, each arc — there is a unique value of sin x, of cos x, and so on. The definition of a function is satisfied. (Topic 3 of Precalculus. ) Moreover, when we draw the graph of y = sin x (Topic 18), we can imagine the unit circle rolled out in both directions onto the x-axis, and in that way marking the coordinates ? , 2? , ?? , ? 2? and so on, on the x-axis. Because radian measure can be identified as an arc, the inverse trigonometric functions have their names. “arcsin” is the arc — the radian measure — whose sine is a certain number. The ratio | sin x x| In the unit circle, the opposite side AB is sin x. sin x| =| AB 1| = AB. | One of the main theorems in calculus concerns the ratio | sin x x| for| very small values of x. And we can see that when the point A on the circumference is very close to C — that is, when the central angle AOC is very, very small — then the opposite side AB will be virtually indistinguishable from the arc length AC.
That is, sin x| | x| | sin x x| | 1. | An angle of 1 radian An angle of 1 radian refers to a central angle whose subtending arc is equal in length to the radius. That is often cited as the definition of radian measure. Yet it remains to be proved that if an arc is equal to the radius in one circle, it will subtend the same central angle as an arc equal to the radius in another circle. We cannot avoid the main theorem. In addition, although it is possible to define an “angle of 1 radian,” does such an angle actually exist? Is it possible to draw one — a curved line equal to a straight line?
Or is that but another example of fantasy mathematics? See First Principles of Euclid’s Elements, Commentary on the Definitions; see in particular that a definition asserts only how a word or a name will be used. It does not assert that what has been defined exists. Problem 1. a) At a central angle of | ? 5| , approximately what ratio has the arc to the| a) radius? Take ? 3. The radian measure | ? 5| is that ratio| . Taking ? 3, then the| arc is approximately three fifths of the radius. b) If the radius is 15 cm, approximately how long is the arc? s = r? 15· | 3 5| = 9 cm|
Problem 2. In a circle whose radius is 4 cm, find the arc length intercepted by each of these angles. Again, take ? 3. a) | ? 4| | s = r? 4· | 3 4| = 3 cm| b) | ? 6| | s = r? 4· | 3 6| = 4· ? = 2 cm| c) | 3? 2| | s = r? 4· | 3· 3 2| = 4· | 9 2| = 2· 9 = 18 cm| d) 2?. (Here, the arc length is the entire circumference! ) s = r? = 4· 2? 4· 6 = 24 cm| Problem 3. In which quadrant of the circle does each angle, measured in radians, fall? (See the figure above. ) a) ? = 2| | 2 radians are more than | ? 2| but less than ?. (See the| figure above. ) Therefore, ? 2 falls in the second quadrant. b) ? = 5| | 5 radians are more than | 3? 2| but less than 2?. (See the| figure above. ) Therefore, ? = 5 falls in the fourth quadrant. c) ? = 14| | 14 radians are more than 2 revolutions, but slightly| less than 2? : 6. 28 + 6. 28 = 12. 56. (See the figure above. ) Therefore, ? = 14 falls in the first quadrant. Proof of the theorem In any circles the same ratio of arc length to radius determines a unique central angle that the arcs subtend; and conversely, equal central angles determine the same ratio of arc length to radius.
Proportionally, if and only if ?1 = ? 2. For, if and only if Now 2? r is the circumference of each circle. And each circumference is an “arc” that subtends four right angles at the center. But in the same circle, arcs have the same ratio to one another as the central angles they subtend. Therefore, and Therefore, according to line (1), if and only if ?1 = ? 2. Therefore, the same ratio of arc length to radius determines a unique central angle that the arcs subtend. Basic Concepts: In Brief, The Sexagesimal System, Centesimal System and the radian measure help in converting the angles.