MA Lesson 19 Summer 2016 Angles and Trigonometric Functions

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1 DEFINITIONS: An angle is defined as the set of points determined by two rays, or half-lines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common point. We call l 1 the initial side, l the terminal side, and O the vertex of angle AOB. The direction and number of rotations of l 1 makes before stopping at l is not restricted. If two angles have the same initial and terminal sides, they are coterminal angles. vertex l terminal side l coterminal angles l 1 l 1 initial side A straight angle is an angle whose sides lie on the same straight line but extend in opposite directions from its vertex. If we introduce the rectangular coordinate system, then the standard position of an angle is obtained by placing the vertex at the origin and letting the initial side l 1 coincide with the positive x-axis. If l 1 is rotated in a counterclockwise direction to the terminal position l, then the angle is considered positive. If l 1 is rotated in a clockwise direction, the angle is negative. A negative angle in standard position A positive angle in standard position An angle is called a quadrantal angle if its terminal side lies on a coordinate axis. One unit of measurement for angle is the degree. The angle is standard position obtained by one complete revolution in the counterclockwise direction has measure 360 degrees, written 360. Two coterminal angles will differ by multiples of 360. To find a coterminal angle, add or subtract multiples of 360. Ex 1: below. Find two positive coterminal angles and two negative coterminal angles for the angles = 55 = 100 1

2 A right angle is half of a straight angle and has measure 90. An acute angle ; 0 < < 90 An obtuse angle ; 90 < < 180 Complementary angles, ; + = 90 Supplementary angles, ; + = 180 Complementary angles have a sum of 90. Supplementary angles have a sum of 180. For smaller units than degrees we have two choices: 1) tenths, hundredths, thousandths, and ten- thousandths of a degree. example: ) divide the degrees into 60 equal parts, called minutes (denoted by ˊ) and each minute into 60 equal parts, called seconds (denoted by ") example: 50 = 49 59' 60" Ex : Find the angle that is complementary Find the angle that is supplementary to = 45.7 to = Find the angle that is complementary. Find the angle that is supplementary to = 39 34' 19" to = 5 13' 45" To convert from a decimal part of a degree to minutes and seconds: Multiply the decimal part of the degree by 60 minutes/degree. The result is the number of minutes. Take the decimal part of a minute and multiply by 60 seconds/minute and round to nearest number of seconds. Multiply by the ratios or. This is DD DMS! 1 1 Ex 3: Express the angle in terms of degrees, minutes and seconds. = =

3 To convert from minutes and seconds to decimal part of a degree: Write x minutes as x/60 of a degree and y seconds as y/3600 of a degree, convert to decimals, add and round. Multiply by the ratios 1 or 1 This is DMS DD Express the angle as a decimal to the nearest ten-thousandth. Ex 4: = 45 16' 45" = ' 1" Another unit of measurement for angles is the radian. One radian is the measure of the central angle of a circle subtended by an arc equal in length to the radius of the circle. Since the circumference of a circle is r, the number of times r units can be laid off around the circle is. ******Thus 360 = radians and therefore 180 = radians. ****** When the radian measure of an angle is used, no units will be indicated. = 5 means = 5 radians. θ = 5 means 5 degrees. Since 180 = radians, use the ratios radians 180 and (since both = 1) to covert degree 180 radians measures to radians and radian measures to degrees. To remember which ones to use, think of which unit needs to be canceled and which unit is to be left as the label. Multiply by 180 to convert degrees to radians. Ex 5: Find the exact radian measure. 180 Multiply by to convert radians to degrees. ex. 30 ex. 60 ex. 90 ex. 10 ex. -45 ex. 50 ex. 450 ex

4 Ex 6: Find the exact degree measure Ex 7: Approximate the number of degrees to 3 decimal places. 3 radians Ex 8: Express the angle in terms of degrees, minutes and seconds. = 3 = 5.6 Remember: Coterminal angles vary by multiples of 360. Since 360 equals π, to find a coterminal angle that is in radian measure, add or subtract multiples of π. Two coterminal angles will differ by multiples of π radians. To find a coterminal angle, add or subtract multiples of π. Ex 9: Find two positive coterminal angles and two negative coterminal angles. = = 7 =

5 Remember, when converting from: degrees to radians: multiply by 180 Ex 10: =.3 radians to degrees: multiply by Express in terms of degrees, minutes and seconds. 180 Find the exact radian measure of. = 40 Find the length of arcs and the area of a sector: If is in radians then the length of the arc created by is found by: s = r (From C = r) C r For a circle: C Therefore: C r which means in general s r Ex 11: Find the length of the arc subtended by the central angle. s s cm cm 5

6 1 If is in radians then the area of the sector created by is found by: A = r (From A = r ) A r Since = for a circle Substitute: A r 1 or A r Ex 1: Find the area of the sectors subtended by the central angle in the examples at the bottom of page 5. Ex 13: Find the radian and degree measures of the central angle subtended by the given arc of length s on a circle of radius r. a) Find the radian and degree measures (DD and DMS) of the central angle. b) Find the area of the sector determined by s = 8 cm, r = 3 cm s = 6 ft, r = 18 in. 6

7 Ex 14: a) Find the length of the arc that subtends the given central angle on a circle of diameter d. b) Find the area of sector determined by = 50, d = 16 m = 3.1, d = 44 cm Ex 15: The minute hand of a clock is 6 inches long. How far does the tip of the minute hand move in 15 minutes? How far does it move in 5 minutes? 7

8 We will introduce the trigonometric functions in the manner in which they originated historically- as ratios of sides of a right triangle. A triangle is a right triangle if one of its angles is a right angle. Opposite Adjacent Hypotenuse If θ would be changed to the other acute angle of the triangle, the opposite and adjacent sides would switch. Labeling triangles is always important. Changing θ changes the adjacent and opposite sides. The hypotenuse is always the same side no matter where θ is put. With as the acute angle of interest, the adjacent side is abbreviated adj., the opposite side is abbreviated opp., and the hypotenuse is abbreviated hyp. With this notation, the six trigonometric functions become: sin opp hyp csc hyp opp cos adj hyp sec hyp adj tan opp adj cot adj opp SOH-CAH-TOA sine is opposite over hypotenuse cosine is adjacent over hypotenuse tangent is opposite over adjacent The sine, cosine, and tangent are the 3 major or primary trigonometric functions. The cosecant, secant, and cotangent functions are the lesser or minor functions. Notice that the csc is the reciprocal of sin, sec is the reciprocal of cos, and cot is the reciprocal of tan. sin 1 csc csc 1 sin cos 1 sec sec 1 cos tan 1 cot cot 1 tan These are called the RECIPROCAL IDENTITIES. Since the hypotenuse is always the largest side of the triangle, 0 sin 1 0 cos <1 Because the hypotenuse is always the (in a right triangle) largest side, any of the functions with csc 1 sec 1 hyp in the denominator, will be less than 1. Any with hyp in the numerator will be greater than 1. 8

9 Ex 16: Find the exact values of the six trigonometric functions for the angle 1 c a Ex 17: Find the exact values of the six trigonometric functions for the acute angle cos 9 41 cot 35 1 csc 5 sec 5 sin 7 3 tan

10 Ex 18: Using your calculator, find (approximate to 4 decimal places): DEGREES sin(134) cos(-54) tan(11) sec(-94) csc(5) cot(330) RADIANS sin(5) cos(-0.13) tan 6 5 sec(3.1) csc(5.6) cot(-9) 10

11 This is the table of trigonometric values you should be able to recall or derive. Reverse the sine row Divide sine row ANGLES θ θ θ Degrees Radians sin cos by cosine row tan This table can easily be verified by examining a right triangle and a right triangle. (See below.) Ex 19: Find the exact values of x and y. A B C 5 y y 8 x y x x 6 11

12 Always draw a picture with applied problems! Ex 0: A building is known to be 500 feet tall. If the angle from where you are standing to the top of the building is 30, how far away from the base of the building are you standing? Ex 1: The angle to the top of a flag pole is 41. If you are standing 100 feet from the base of the flagpole, how tall is the flagpole? Ex : The angle to the top of a cliff is If you are standing 300 meters from a point directly below the cliff, how high is the cliff? 1

13 The Fundamental Identities: (1) The reciprocal identities: csc 1 sin sec 1 cos () The tangent and cotangent identities: tan sin cot cos cos sin cot 1 tan (3) The Pythagorean identities: sin cos 1 1 tan sec 1 cot csc Note: (sin ) is usually written sin. Similar notation for the other functions. The reciprocal identities are obvious from the definitions of the six trigonometric functions. Take the simple right triangle with sides 3, 4, and 5 with θ opposite the side of length sin csc cos sec tan cot θ To prove the tangent identity, examine the following. The cotangent identity proof is similar opp sin cos tan opp opp hyp hyp sin tan 3 adj hyp adj adj cos sin hyp cos sin tan cos Use the simple right triangle with sides 3, 4, and 5 on page 13 to find sin, cos, and sin cos. This information can be used to prove the Pythagorean Identities. See the proofs on the next page. sin cos is called a Pythagorean identity since it is derived from the Pythagorean Theorem. 13

14 3 4 sin cos 5 5 (sin ) (cos )? sin cos? 3 4? ? sin cos 1 x y r opp adj hyp x y r opp adj hyp r r r hyp hyp hyp cos sin 1 sin cos 1 Divide both sides of the Pythagorean identity above by identity. sin cos 1 Divide each side by sin sin cos 1 sin sin sin cos 1 csc sin 1 cot csc sin to get a second Pythagorean To get the remaining Pythagorean identity, divide each side of the identity cos. sin cos 1 by Each of the three Pythagorean identities creates two more identities by subtracting a term from the left side to the right side. sin cos 1 1tan sec 1cot csc sin 1cos cos 1sin tan sec 1 cot csc 1 1 sec tan 1 csc cot 14

15 Ex 3: Verify each identity by transforming the left side to the right side. (a) tan cot 1 ( b) sin(3 )cot(3 ) cos(3 ) sin cos sec ( c) csc ( d) 1 cot tan sin ( e) (1 cos )(1 cos ) sin ( ) cos (sec 1) sin f 15

16 ( g) (1 sin )(1 tan ) 1 ( h) cot tan csc sec Below is a picture of an angle θ drawn in standard position (vertex at the origin and the initial side on the positive x-axis. (x, y) y= opposite x = adjacent Ex 4: If θ is an angle is standard position on a rectangular coordinate system and if P( 5,1) is on the terminal side of θ, find the values of the six trigonometric function of θ. (Hint: Draw a picture.) 16

17 Ex 5: If θ is an angle in standard position on a rectangular coordinate system and if P (4,3) is on the terminal side of θ, find the values of the six trigonometric functions of θ. Ex 6: Find the exact values of the six trigonometric functions of θ, if θ is in standard position and the terminal side of θ is in quadrant III and on the line with equation 4x3y 0. Notice: tan slope of the line Ex 7: Find the exact values of the six trigonometric functions of θ, if θ is in standard position and the terminal side of θ is in quadrant II and parallel to the line with equation 3x y7 0 17

18 Ex 8: Find the quadrant containing θ if the given conditions are true. a) tan < 0 and cos > 0 b) sec > 0 and tan < 0 c) csc > 0 and cot < 0 d) cos < 0 and csc < 0 II III sin + csc tan cot + All + cos sec + I IV e) cos θ < 0 and sec θ > 0 To help you remember the picture above: Think (in order of quadrants): ALL STUDENTS TAKE CALCULUS. Quadrant I: All functions are positive values. Quadrant II: Sine and its inverse are positive, others negative. Quadrant III: Tangent and its inverse are positive, others negative. Quadrant IV: Cosine and its inverse are positive, other negative. Ex 9: Use the fundamental identities to find the values of the trigonometric functions for the given conditions below (a) 1 tan and cos 0 5 (b) sec 4 and csc 0 18