MATH 181-Trigonometric Functions (10)
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1 The Trigonometric Functions ***** I. Definitions MATH 8-Trigonometric Functions (0 A. Angle: It is generated by rotating a ray about its fixed endpoint from an initial position to a terminal position. The fixed endpoint is the vertex. ex. B. Counterclockwise rotation: The rotation of the terminal side from the initial side;it is positive. ex. C. Clockwise rotation: Rotation of counterclockwise rotation; it is negative. ex. D. Degrees: Units of measurement of an angle;a degree is of one 60 complete rotation. (Symbol for degree is. E. Minutes:Units of measurement of an angle; a minute is of a degree. 60 (Symbol for minute is. F. Seconds:Units of measurement of an angle; a second is of a minute. 60 (Symbol for second is. G.Coterminal angles: Two angles sharing the same initial and terminal II. Angle Conversions sides. ex. A. Conversion between radians and degrees
2 . Radian, another measure of an angle, is defined in terms of an arc of a circle. rad = 80 (such that rad is the symbol for radian. A radian is the measure of an angle with its vertex at thte center of a circle whose intercepted arc is equal to the length of the radius of the circle. 80. From rad to degree: multiply by the factor ( 80 ex. Express rad in degrees: rad = ( Express 6.8 rad in degrees: 6.8 ( 60. From degree to rad: multiply by the factor ( 80 ex. Express 80 in rad: 80 ( 80 Express 60 in rad: 60 ( 80 rad Express 4.6 in rad: 4.6 ( 80 rad rad 4. From degrees/minutes (/' to decimal form, knowing 60': # of + # of ' multiplied by ( 60' ex. Express 80 in decimal form: 8+0 ( 60' = From decimal to degrees/minutes ( /' : ex. Express 8.5 in degrees/minutes ( /' form: 8 and 0.5( 60' 0 ' 80 B. Standard Position of an Angle
3 . Standard position:angle is in this position if the initial side of the angle is the positive x-axis and the vertex is the origin. The angle is then determined by the terminal side. An angle is in standard position when its vertex is located at the origin and its initial side is lying on the positive x - axis. A positive angle is measured CCW;a negative angle is measured CW.. First-quadrant angle: angle with terminal side in the firstquadrant. (And so with second-quadrant or third-quadrant angle. *Hint: angle between o 90 : first-quadrant angle angle between : second-quadrant angle angle between 80 angle between : third-quadrant angle 60 :fourth-quadrant angle. Quadrantal angle: angle in standard position with its terminal side coinciding with one of the coordinate axes. (ex.standard position angle of 90, 80, 70, 60 ***** Defining the Trigonometric Functions I. Quick Review of Similar Triangles A. Corresponding angles are equal. B. Corresponding sides are proportional. C. Corresponding sides are the sides, one in each triangle, which are between the same pair of equal corresponding angles. D. For two similar triangles, the ratio of one side to another side in one triangle is the same as the ratio of the corresponding sides in the other triangle.
4 ex. II. Trigonometric Functions Due to similar triangle property D above, for any angle in standard position, six ratios may be established. The values of these ratios depend on the size of the angle, and there is only one value for each ratio for a given angle. Since these ratios are functions of the angle, we call them the trigonometric functions. sine of = hypoteneuse secant of = hypoteneuse adjacent cosine of = adjacent hypoteneuse cosecant of = hypoteneuse tangent of = adjacent cotangent of = adjacent * Do not forget the Pythagorean Theorem: c a b (with hypoteneuse c And legs a and b. ex. Determine the trigonometric functions of the angle whose terminal side passes through ( 6, 8. Firstly, find the length of the hypoteneuse using Pythagorean Theorem. hypoteneuse sine = hypoteneuse = cosine = adjacent hypoteneuse 8 hypoteneuse 0 tangent. secant 67. adjacent 6 adjacent cosecant hypoteneuse = 0 8 adjacent 6 5. cotangent
5 sine = sin cosine = cos tangent = tan secant = sec cosecant = csc cotangent = cot *In many texts, the hypoteneuse is also being referred to as radius vector. III. Trigonometric Functions Values *Be able to calculate function values if one point on the terminal side of the angle is known. *Be able to calculate the trigonometric functions of angles in degrees. ex. In a triangle, the side 0 angle is half of the hypoteneuse. If using = side and = hypoteneuse, then adjacent side = sin0 hypoteneuse cos0 adjacent hypoteneuse tan0 adjacent sec0 hypoteneuse adjacent csc0 hypoteneuse cot0 adjacent sin cos tan
6 *Inverse Trigonometric Function: sin x (or arcsin x means for angle whose sine is x ex. If sin0, then sin 0 If cos0, then cos 0 If tan 45, then tan 45 *Be sure you know how to use your graphing calculator to find these trigonometric function values and inverse trigonometric function values. ***** Signs of the Trigonometric Functions I. Recalling definitions of the trigonometric functions, using the figure below: Point ( x, y is on terminal side of angle. r is the radius vector. sin y r cos x r tan y x csc r y sec r x cot x y II. Relationship among x, y, and r A. r is always positive. B. Signs of trigonometric functions depend on signs of x and/or y. C. For each trigonometric function:. sin, based on y, quadrant: I (+, II (+, III (-, IV (- r 6
7 . cos, based on x, quadrant: I (+, II (-, III (-, IV (+ r. tan, based on y, quadrant: I (+, II (-, III (+, IV (- x 4. csc, based on r, quadrant: I (+, II (+, III (-, IV (- y 5. sec, based on r, quadrant: I (+, II (-, III (-, IV (+ x 6. cot, based on x, quadrant: I (+, II (-, III (+, IV (- y III. Determine the signs of the given trigonometric functions Hint: draw a diagram A. sin50 N. cos( 0 B. sin( 0 O. csc( 70 C. tan95 P. sec7 D. sec5 Q. csc5 E. sin( 40 R. cot( 00 F. cos( 70 S. sec50 G. tan6 T. tan70 H. cot( 5 U. csc70 I. cos56 V. cot65 J. sin0 W. sec( 65 K. cos70 X. csc( 80 L. tan6 Y. sin( 70 M. cot( 5 Z. sin70 7
8 ***** IV. Find the trigonometric functions of if the terminal side of passes Through: (, 4 ; (-, 6 ; (-6, -5 ; (4, -6 V. Determine the quadrant in which the terminal side of lies, with given Conditions: A. tan positive, cos negative B. sec negative, cot negative C. cos positive, csc negative D. sec positive, csc positive E. cot negative, sin negative Trigonometric Functions of Any Angle I. The trigonometric functions of coterminal angles are the same. Try: trig functions of following coterminal angle pairs: Ex. A. 50 and -0 B. 90 and -70 C. 60 and 880 D. -60 and 00 II. The values of the functions depend only on the values of x, y, and r. A. The absolute values of a function of a second-quadrant angle is equal to the value of the same function of a first-quadrant angle. B. If, triangles containing angles and are congruent. If is supplementary to and, 8
9 F( F( F(, or F( F( 80 F( F( F( 80 F( F( F( 60 F( 4 4 such that F refers to any of the trigonometric functions. C. The reference angle of a given nonquadrantal angle is the acute angle formed by the terminal side of the angle and the x-axis. D. The sign depends on whether the function is positive or negative in the second/third/fourth quadrant. E. Examples: Express the given trigonometric function in terms of the same function of a positive acute angle. *Keep in mind that, using as the reference angle and as the required angle: *Find the reference angle first. First Quadrant: Second Quadrant: 80 Third Quadrant: 80 Fourth Quadrant: 60. tan 95 cot( 8. csc( sec( 0 5. csc( sin( 0 7. sin( 0 8. tan( 0 9. cos( 0 0. cot( 7. tan( 0. csc( 0. sec55 4. sec( 7 5. cos90 6. cot( 0 9
10 7. sin97 8. tan( sec( 5 0. cot( 00. csc( 450. sec( 0. csc( sin( 0 5. sin( 0 6. cos( cos( 0 8. cot( 7 9. tan( 0 0. csc( 0 F. A quadrantal angle is the angle for which the terminal side is along one of the axes. Keep in mind that r is always positive. sin cos tan cot sec csc undef..000 undef undef undef undef undef undef undef undef..000 y 0, x x 0, y x r, y x 0, y r 0
11 sin cos cos sin *Note: If and are complementary angles, tan cot cot tan sec csc csc sec *Inverse Trigonometric Function: sin x (or arcsin x means for angle whose sine is x ex. If sin0, then sin 0 If cos0, then cos 0 If tan 45, then tan 45 G. Find the trigonometric function values by finding the reference angle and attaching the proper sign. Then check against calculator s values.. sin85. cos 6.. tan0 4. sec40 5. csc60 6. cos40 7. cot46 8. csc( 0
12 H. Find, for sin cos tan cot csc sec tan 09.,sin 0 8. cos 0576.,sin 0 9. sin 09., tan 0 0. cot 056.,sin 0. Find sin when cos 0. 4 and tan 0.. Find cot when sec. and sin < 0.
13 Radians Arbitrary division/definition of a circle into 60 parts was influenced by ancient Babylonians use of number system based on 60 rather than 0. I. A radian is the measure of an angle with its vertex at the center of a circle and with an intercepted arc on the circle equal in length to the radius of the Circle. 60 Prad or rad = 80 or = 90 II. Conversion between rad and degree A. From degrees to rad.: multiply by ( /80 B. From rad to degrees: multiply by (80/ III. Special note concerning radian measure A. Radian measure amounts to measuring angles in terms of real numbers. B. When no units are indicated, the radian is understood to be the unit of angle measurement. C. Be careful to have the calculator in the proper mode. IV. Examples A. The velocity v of an object undergoing simple harmonic motion at the end of a spring is given by: v A k m cos k m t k The angle is ( ( t. Be sure to use the radian mode to find the m cos k m t. Mass of the object is m (in grams. The spring constant is k. The maximum distance the object moves is A Find the velocity (in centimeters per second after 0.00 s. of a 40-g
14 object at the end of a spring for which k 400g / s, if A 400. cm. B. Using quadrantal angles to estimate which quadrant the angle is in. Degrees Radians Radians(decimal Angle of 5.45 is in fourth quadrant and the reference angle = 0.98 C. More example problems. Express the given angle measurements in radian measures in terms of. a.. 75 b. 0 c. d. 0 e. 60 f. 50. Express each angle in terms of degrees. a. b. 8 c. 4 5 d. 5 4 e. 6 f. 5. Evaluate the given trigonometric functions by first changing the radian measure to degree measure. Round off results to four significant digits. a. sin 4 b. cos 6 c. tan 4 d. cot.7 e. sec
15 4. Find to four significant digits for 0 ( = 6.8. a.. sin b. tan 06. C. cos ***** Applications of the Use of Radian Measure I. Arc Length A. The length of an arc on a circle is proportional to the central angle. B. The length of arc of a complete circle is the circumference. s r (s tands for length of arc and is the central angle in radians of a complete circle. C. The length of any arc: s r ( in radians D. Ex. If 50, and r 400. in. Find the length of this arc. s 50 ( 400. in. 05. in. 80 II. Area of a Sector of a Circle A. Areas of sectors of circles are proportional to their central angles. B. Area of a circle is A r (which is also A r ( C. Therefore, the area of any sector of a circle in terms of the radius and the central angle (in radians is: A r D. Ex. The area of a sector of a circle with central angle 40, and r 500. in Is: A in in 40 ( ( (
16 III. Angular Velocity A. Average velocity: v s t (s is distance traveled, t is the lapsed time B. If an object is moving around a circular path with constant speed, the actual distance traveled is the length of the arc traversed. s r t t t r is the angular velocity, designated by. t t is the angular displacement in radians. is the angular speed in rad/s. t i s the time in s. v r Defines the relationship of linear velocity v and angular velocity of an object moving around a circle of radius r. C. v is directed tangent to the circle and its direction changes constantly. D. are radians per unit of time. is often given in revolutions per unit of time. E. Ex. An airplane propeller blade is.80 ft. long and rotates at 00 rev/min. What is the linear velocity of a point on the tip of the blade? Use v r t r rev rad 00 ( ( 40. ft 9400 ft rev F. revolution = radians. 6
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