MATHEMATICS 105 Plane Trigonometry

Chapter I THE TRIGONOMETRIC FUNCTIONS MATHEMATICS 105 Plane Trigonometry INTRODUCTION The word trigonometry literally means triangle measurement. It is concerned with the measurement of the parts, sides, and angles of a triangle. It further deals with six ratios that are determined by angles and directed line segments. The main purpose of the study of trigonometry is to solve problems involving triangles as what are found in astronomy, navigation and surveying. With the development of trigonometry, trigonometric functions are associated with the lengths of the arcs on the unit circle. It is now defined as a branch of Mathematics which is concerned with the properties and applications of circular or trigonometric functions. Plane Trigonometry, which is the concentration of this course, is restricted to the study of triangles lying in a plane. DIRECTED LINE SEGMENTS Two concepts that are associated with a line segment are distance and direction. The distance from one point to another, or the length of the line segment between the two points, is the number of times that an accepted unit can be laid off along the given segment. When a line segment is measured with a definite sense or direction from one endpoint to the other, the segment is said to be directed line segment. The distance between two distinct points on a directed line segment is called a directed distance. A fundamental property of directed line is that if P1, P and P are any distinct points on the line, then the following relation holds: P 1 P P 1 P + P P On line L, we can assign an origin, a unit length and a positive direction. Figure 1, illustrates a line where the distance between 0 and 1 corresponds to the unit length, the arrow at the right indicates positive direction. This line is a one-dimensional coordinate system or a number line, where a one-to-one correspondence can be established. - - -1 0 1 The directed distance from P1(x1) to P(x) regardless of the relative positions of 0, P1 and P is given by P 1 P x - x 1 The undirected distance between P1 and P is the absolute vale of the directed distance between them. P 1 P x - x 1 Find the directed distance from P1 to P given the following: No. P 1 P No. P 1 P 1 6-4 6.580 8.098 k k 7 ¾ 5 ½ 5 ¾ 8.7-5. 4 7-8 9 7 ½ 9. 5 -b a 10-1.0007.08 1

50 m 14 m

THE RECTANGULAR COORDINATE SYSTEM Points in a plane are located using coordinate axes which consists two perpendicular lines X OX (horizontal) and Y OY (vertical) which intersect at the point O (refer to the figure at the right). The line X OX is called the x-axis and the line Y OY is called the y-axis. Together they are known as the Cartesian coordinate axes. The point of intersection is called the origin. The coordinate axes divide the plane into four distinct regions called quadrants marked I, II, III, IV (labelled counter clockwise). In quadrant I, both x and y coordinates are positive (x > 0, y > 0), in quadrant II, x- coordinate is negative, y-coordinate is positive (x < 0, y > 0); in quadrant III, both are negative (x < 0, y < 0) and in quadrant IV, x- coordinate is positive, y-coordinate is negative, (x > 0, y < 0). The Cartesian Coordinate System Y (x < 0, y > 0) 5 (x > 0, y > 0) II 4 P(x,y) I 1 4 5 O X' -5-4 - - -1 X III - - IV (x < 0, y < 0) -4-5 (x > 0, y < 0) Y' 1. Plot the points whose coordinates are: a. P1 (5, 4) b. P (4, -1) c. P (0, ) d. P4 (-, -5). Give the coordinates of the following points and the quadrants where they are located: a. five units to the right of the y-axis and two units below the x-axis b. two units to the left of the y-axis and one unit below the x-axis THE DISTANCE FORMULA The distance between two points P 1 and P can be expressed in terms of their coordinates by the Pythagorean theorem: Let the coordinates of two points be denoted by P 1 (x 1, y 1 ) and P (x, y ), by Pythagorean theorem, Y P 1 (x 1,y 1 ) x 1 x d (x - x 1 ) P (x,y ) (y - y 1 ) y 1 y X d ( x y x1 ) + ( y 1) 1. Find the distance between two points: a. P1(5, 14) P(-10, ) b. P1(, 5) P(-1, ) If the two points in the distance formula are the origin and the point whose radius vector r we want, we find that 0 0. Consequently, squaring each member of this equation, a relation between the coordinates and the radius vector r of P(x,y) is. 1. If the abscissa of a point is 1 and its radius vector is 1, find the values of the ordinate. Solution: x 1, r 1, y? 1 1 144 169 169 144 5 5. If the radius vector of a point in the second quadrant is 5 and the ordinate is, find the abscissa. Solution: x?, r 5, y 5 9 5 16 4 Since the point is in the second quadrant, we will use x - 4

1. Plot each point whose coordinates are given and find the radius vector of each. Furthermore, find the distance between each pair of points. A(, ) B(7, - 5) C(-8, 4) D(-, -6). Find the value or values of the one of x, y and r that is missing in each of the following: a. x, y 4 b. x -4 r 5 c. x 5 y 11 d. x - r 5, y < 0 e. y r 7 x < 0 ANGLES Angles are what trigonometry is all about. This is where it all started, way back when. Early astronomers needed a measure to tell something meaningful about the sun and moon and stars and their relationship between man standing on the earth or how they are positioned in relation to one another. Angles are the input values for trigonometric functions. An angle is formed where two rays (straight line with an endpoint that extends infinitely in one direction) have a common endpoint. This endpoint is called the vertex. The two rays are called the sides of an angle initial and terminal sides. A plane angle is to be thought of as generated by a revolving (in a plane) a ray from the initial position to a terminal position. In Figure 1, NOP or θ, ray OP OP is the initial side, while ray ON ON is the terminal side. Writing Angle Names Correctly. An angle can be identified in several different ways: Use the letter at the vertex of the angle. Use the three letters that label the points one on one side, the vertex and the last on the other ray. Points are labelled with capital letters. Use the letter or number in the inside of the angle. Usually, the letters used are Greek or lowercase. Give all the different names that can be used to identify the angle shown in Figure. Measure of an angle. An angle, so generated, is called positive if the direction of rotation is counterclockwise and negative if the direction of rotation is clockwise. The common unit of measure of an angle is degree denoted by ( ). Classification of Angles. Angles can be classified by their size. Acute Angle an angle measuring less than 90. Right Angle an angle measuring exactly 90 ; the two sides are perpendicular Obtuse Angle an angle measuring greater than 90 and less than 180. Straight Angle angle measuring exactly 180. Two angles can also be classified according to the sum of their measures. If the sum of the measures of the angles is 90, then the angles are called complimentary angles. If the sum is 180, then the angles are called supplementary angles. 4

When two line cross one another, four angles are formed. These angles which are opposite one another are called vertical angles. If two angles are vertical, then their measures are equal. 1. If one angle in a pair of supplementary angles measure 80, what does the other angle measure? Answer: The other measures 180-80 100. What is the measure of the (a) complement (b) supplement of the angle in Figure 1 if θ is 8? Answer: (a) complement is 90-8 5 (b) supplement is 180-8 14 1. Find the measure of the complement of an angle whose measure is: a. 7 b. 58 c. x d. 90 - y e. 7 + R. Determine the measure of the supplement of an angle whose measure is: a. 75 b. 85 c. x d. 90 + y e. 11.76 + R ANGLE IN STANDARD POSITIO OSITION An angle is in standard position with reference to a system of rectangular coordinate axes if its vertex is at the origin and its initial side lines along the positive ray of the X axis. If an angle is in standard position, the angle is said to be in the quadrant in which the terminal side lies. Hence, an acute angle is in the first quadrant; an obtuse angle is in the second quadrant; an angle of 15 is in the third quadrant; an angle of 0 is in the fourth quadrant. If the terminal side of an angle in standard position coincides with one of the coordinate axes, the angle is a quadrantal angle. An angle of 90 and any angle which is an integral multiple of 90 is a quadrantal angle. Two angles are coterminal if they are in standard position and have the same terminal sides. Thus, of 150, 510, and - 10 are coterminal angles. 1. Construct the following angles in standard position and determine those which are coterminal: a. 15 f. -70 b. 10 g. -955 c. -150 h. -870 d. 85 i. 595 e. 90 j. -10. Give three other angles coterminal with a. 15. b. 45 c. 165 d. 590 e. -0 f. 75 g. -10 h. -60 i. 70 j. 15 5

DEFINITION OF THE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE IN STANDARD POSITION The six ratios on which the subject of trigonometry is based are formed by using the abscissa, the ordinate, and the radius vector of a point on the terminal side of an angle in standard position. Since the values of these ratios depend on the values of the abscissa, the ordinate, and the radius vector, and these in turn depend on the size of the angle, each ratio is a function of the angle. We shall now give the definitions of these six functions and, beside each definition, the abbreviated form in which it is usually written. These definitions and abbreviated forms should be memorized. If the angle θ is in standard position, then The sine of angle θ sin θ The cosecant of angle θ csc θ The cosine of angle θ cos θ The secant of angle θ sec θ The tangent of angle θ tan θ The cotangent of angle θ cot θ It should be noted that the cosecant is the reciprocal of the sine, the secant the reciprocal of the cosine, and the cotangent the reciprocal of the tangent. The values of the functions of an angle are not affected by the position of the point P on the terminal sides as can seen by the following consideration. If P and P are any two points on the terminal side of θ, and if QP and Q P are perpendicular to the X axis, then the triangles OQP and OQ P are similar; hence, corresponding sides are proportional. Thus,, and we see that the sine ratio has the same value regardless of the point on the terminal side that is used. In a similar way it can be shown that the values of the other five functions are not affected by the position of P. 1. If the point (5,-1) is on the terminal sides of an angle θ that is in standard position, find sin θ, cos θ, and tan θ. Solution: By use of the relation between the abscissa, ordinate, and radius vector of a point, we see that 5 1 5 144 169. Hence, the radius vector is 1, and we have sin θ, cos θ, tan θ. 1. Determine the values of the trigonometric functions of angle θ if P is a point on the terminal side of θ and the coordinates of P are: a. P(, 4) b. P(-, 4) c. P(-1, -) d. P(, 1 e. P(, -) 6

ALGEBRAIC SIGNS OF THE TRIGONOMETRIC FUNCTIONS The algebraic signs of the functions of an angle depend on the signs of the abscissa and the ordinate, since the radius vector is always positive. For example, since the abscissa of every point in the second quadrant is negative and the ordinate is positive, any function which uses the abscissa is negative while all others are positive. However, for any point in the third quadrant, the abscissa and the ordinate are both negative; hence, the cotangent are positive since each is the ratio of two negative numbers. The other functions are negative. Figure at the right shows functions are positive for each quadrant; all other functions are negative. FUNCTIONS OF 0 0, 45 45 AND THEIR MULTIPLES The computation of the values of trigonometric functions of angles in general is beyond the scope of an elementary trigonometry book. We can, however, find the values of functions of 0, 45, 60, and their multiples by use of some theorems from geometry and the definition of the trigonometric functions. We shall first consider a 45 angle in standard position as shown in Fig. 1-11. From plane geometry, we know that a 45 right triangles is isosceles. If each of the equal sides is 1 unit long, by the Pythagorean Theorem the hypotenuse is units. Hence, the coordinates and radius vector P are as shown in Fig. 1-11. The definition of the trigonometric functions gives Sin 45, cot 45 1, cos 45, tan 45 1, Sec 45, csc 45, In order to find the functions of 0 and 60, we use the theorem from plane geometry which states that, in a 0 right triangle, the side opposite the 0 angle is half as long as the hypotenuse. Hence, if we put the 0 angle in standard position, the coordinates and the radius vector of P are as shown in Fig. 1-1, since the Pythagorean theorem gives x 1 and x. Now by use of the definition, we have Sin 0, cos 0 cot 0, Sec 0, tan 0, csc 0. In other to find the values of the functions of a 60 in standard position as shown in Fig. 1-1. The other acute angle of the 60 right triangles is 0. hence, the values of the coordinates and radius vector of P are as shown. We need now only apply the definitions to see that Sin 60, cos 60 cot 60, Sec 60, tan 60, csc 60. The values of the trigonometric functions of integral multiples of 0, 45 can be found by making use of the values of the functions of 0, 45, and 60 and the reference angle. If an angle is in standard position, then the acute angle between the terminal side and the X axis is called the reference angle. Thus, the reference angle of 150 is 0 and so is the reference of 10. The reference angle of 5 is 45 and that 00 is 60. 7

EXAMPLE 1. Find the values of the six trigonometric functions of 150. Solution Draw an angle of 150 in standard position. The angle formed by the terminal side of the 150 angle and the negative ray of the X axis is the supplement of 150, or 180-150 0. sin 150, csc 150. cos 150 sec 150, tan 150. Evaluate sin 5 - tan 15 Solution:, cot 150, sin 5 - tan 15 - (-1) ( ) + 1 FUNCTIONS OF QUADRANT UADRANTAL AL ANGLES A quadrantal angle was defined as an angle that is in standard position and has its terminal side along a coordinate axis. It follows that the abscissa or the ordinate of a point on the terminal side is zero, and as result has to be used in finding some of the values of the functions of the angle. It must be remembered that zero divided by any number is zero, and that no number can be divided by any number is zero, and that no number can be divided by zero. It follows from these two facts that two of the function of a quadrantal angle are zero and that there is no number to represent another two of the functions. The functions of 90 can be obtained from Fig. 1-16 and are given below sin 90 1, cot 90 0, cos 90 0, sec 90 undefined, tan 90 undefined, csc 90 1. Functions of other quadrantal angles can be found in similar manner. VALUES OF TRIGONOMETRIC FUNCTIONS FOR SOME FAMILIAR ANGLES Angle α Degree Radian sin α cos α tan α csc α sec α cot α 0 0 0 1 0 1 1 0 6 45 1 1 4 1 60 90 1 0 1 0 180 0-1 0-1 70-1 0-1 0 60 0 1 0 1 8

1. Find the functions of the following: a. 0, 15, 40 b. 45, 10, 0 c. 60, 15, 10 d. 00, 5, 150. Evaluate the combination of functions of angles given in each of the following. a. sin 90 cos 0 + cos 90 sin0 c. b. cos 180 cos 60 + sin 180 sin 60 d.. Verify the following statements: a. cos 45 - sin 15 cos 90 b. sin 60 + cos 40 tan 5 c. sin 0 + cos 150 + tan 60 sec 00 d. sec 10 - tan 0 sin 150 + cos 0 e. sin 40 sin 60 cos 40 4. Identify each of the following as true or false. a. sin 150 cos 0 tan 10 b. sin 10 sin 10 cos 00 c. cos 00 cos 0 - sin 0 d. 4 f. cos 150 cos 45 cos 0 - sin 15 sin0 tan 15 g. cot 0 - csc 150 sin 10 cos 00 - h. tan 150 tan 10 1 tan 10 e. tan 150 tan 0 f. cos 90 cos 45 sin 15 g. tan 00 cot 10 tan 5 RELATIONS BETWEEN DEGREES AND RADIANS A degree ( ) is defined as the measure of the central angle subtended by an arc of a circle equal to circumference of the circle. The degree is divided into 60 minutes and a minute is divided into 60 seconds. A minute ( ) is of a degree; a second ( ) is of a minute or of degree. 1. Express each angle measures in degrees, minutes and seconds. a. 18.5 18 + 0.5 18 + 0.5 18 0 b. 6 8 6 + 8.5 6 + 8 +.5 6 8 15 c. 75.15 75 + 0.1 + 0.005 75 + 0.15 75 + 7 + 0.5 75 7 0 75 + 7.5. Express each angle measures in decimal degrees: a. 45 1 6 45 + 1 1 1 + 6 45 + 0. + 0.01 45.1 60 600 b. 86 15.6 86 + 15.6 1 86 + 0.6 86.6 60 A radian (rad) is defined as the measure of the central angle subtended by an arc of a circle equal to the radius of the circle. The circumference of a circle (radius) and subtends an angle of 60. radians 60 radian 180 1 57.96 57 17 45" of the 1 degree 0.01745 9

1. Convert the following into radian. Express answer as multiple of π. a. 0 0. Express the following in terms of degrees: a. 45 b. 10 1. Express the following in degrees, minutes and seconds: a. 6.4 (6 4 ) b..9 ( 54 ) c. 9. (9 1 48 ). Express the following in degrees (rounded to hundredths): a. 78 17 (78.8 ) b. 58 16 (58.7 ) c. 10 0 45 (10.51 ) b. 40 0 40.5 d. 7.47 (7 8 1 ) e. 4.15 (4 7 0 ) d. 5 right angles (450 ) e. 8.5 revolutions (060 ). Convert the following into degree measure: a. b. c. (105 ) (10 ) (5 ) d. 0.10 (1 1 0 ) e. (75 0 ) 4. Convert the following into radian measure. Express answer as multiple of : a. 90 d. 150 b. 16 e. 5 7 0 c. 7 1 5. Express each angle in terms of radians rounded to hundredths. a. 8 d. 44 44 44 b. 4 1 e. 71.71 c. 8 19 0 Assignment: Ref. Book: Plane Trigonometry by Barcelon, et al. p. 9 A to D. 10