Math 4 part 1

Mathematics IV PART 1

Module 1Circular Functions and Trigonometry What this module is about This module is about the unit circle. From this module you will learn thetrigonometric definition of an angle, angle measurement, converting degreemeasure to radian and vice versa. The lessons were presented in a very simpleway so it will be easy for you to understand and be able to solve problems alonewithout difficulty. Treat the lesson with fun and take time to go back if you thinkyou are at a loss. What you are expected to learnThis module is designed for you to: 1. define a unit circle, arc length, coterminal and reference angles. 2. convert degree measure to radian and radian to degree. 3. visualize rotations along the unit circle and relate these to angle measures. 4. illustrate angles in standard position, coterminal angles and reference angles.How much do you knowA. Write the letter of the correct answer.1. What is the circumference of a circle in terms of π ?a. π b. 2π c. 3π d. 4π2. An acute angle between the terminal side and the x-axis is called ______a. coterminal b. reference c. quadrantal d. right

3. 60 ° in radian measure is equal toa. π b. π c. π d. π 2 3 4 64. 2.5 rad express to the nearest seconds is equal toa.143° 14′ 24″ b. 143° 14′ 26″ c. 43° 14′ 26″ d. 43° 14′ 27″5. What is the measure of an angle subtended by an arc that is 7 cm if the radius of the circle is 5 cm?a. 1. 4 rad b. 1.5 rad c. 1.6 rad d. 1.7 rad6. Point M ( 24 , y) lie on the unit circle and M is in Q II. What is the value of 25y?a. 6 b. − 6 c. 7 d. − 7 25 25 25 257. What is the measure of the reference angle of a 315o angle?a. 45 o b. 15 o c. -45 o d. -15 o8. In which quadrant does the terminal side of 5π lie? 6a. I b. II c. III d. IV9. A unit circle is divided into 10 congruent arcs. What is the length of each arc?a. π b. π c. 2π d. 10π 10 5 5B. Solve:10. The minute hand of the clock is 12 cm long. Find the length of the arc traced by the minute hand as it moved from its position at 3:00 to 3:40. 2

What you will do Lesson 1 The Unit Circle A unit circle is defined as a circle whose radius is equal to one unit andwhose center is at the origin. Every point on the unit circle satisfies the equationx2 + y2 + 1.The figure below shows a circle with radius equal to 1 unit. If thecircumference of a circle is defined bythe formula c = 2πr and r = 1, thenc = 2π or 360° or 1 revolution. r =1If 2π = 360° , then π = 180° or ºone-half revolution. Now, imagine the Quezon Memorial Circle as a venue for morningjoggers. The maintainers have placed stopping points where they could relax. If each jogger starts at C BPoint A, the distance he would Atravel at each terminal pointis shown in table below. DStopping B C D A PointDistance or π π 3π 2π Arclength 2 2 3

divided into 4 equal stopping pointsUnits of Angle MeasuresThere are two unit of angle measure that are commonly used:1. Degree measure2. Radian measure. A complete revolution is divided into 360 equal parts. Degree issubdivided to minutes and seconds.1 rev = 360°1° = 60′ ′ is the symbol for minutes1′ = 60″ ″ is the symbol for secondsExample 2.Change each angle measure in decimal to minutes and seconds. a. 54. 5° b. 28. 42°Solution Angle in tenths will be changed to the nearest minutes and angles inhundredths will be changed to the nearest second. a. 54.5° = 54 + 0.5( 60′) = 54° + 3′ = 54° 3′ b. 28. 42° = 28° + 0.42( 60′ ) = 28° + 2.52 = 28° + 2′ + 0.52( 60″ ) 4

= 28° + 2′ + 3.12″= 28° 2′ 3″ A radian is defined as the measure of an angle intercepting an arc whoselength is equal to the radius of the circle. An arc length is the distance betweentwo points along a circle expressed in linear units. arc length θ= sangle in radian = ________________ or rradius of the circle )) For all circles, the radian measure of the circumference is 2πradians. But the angle has a measure of 360°. hence, 2π rad = 360° π rad = 180° 1 rad = 180 or 57.296° π 1° = π rad or 0.017453 rad 180Converting Angle Measures To convert from degrees to radians, multiply thenumber of degrees by π . Then simplify. 180 To convert from radians to degree, multiply thenumber of radians by 180 . Then simplify. πExample 3: 5

Convert the measure of the following angles from degrees to radians. a. 70° b. -225° c. 125° 15′ 45″Solution a. 70° = 70° x π = 7π rad 180 18 b. -225° = -225° x π = − 5π rad 180 4 c. 125° 15′ 45″ 900″ + 45″ = 945″ Convert 15′ to seconds 60″ 15′ x 1′ = 900″ Convert 945″ to degrees 945″ = 0.263° 125° + 0.263° = 125 .263° 3600″ Since, 1° = π rad or 0.017453 rad 180 Therefore, 125 .263° x 0.017453 rad = 2.186 rad 6

Example 4. Express each radian measure in degrees a. 2π b. 2.5 rad to the nearest seconds 4Solution a. 2π x 180 = 90° 4π 2.5 rad x 57.296° = 143.24° .24° x 60′ = 14.4′ 1° .4′ x 60″ = 24″ 1′ ∴ 2.5 rad = 143° 14′ 24″ Example 5. Solve:slipping and a. A wheel of radius 80 cm rolls along the ground withoutwheel move? rotates through an angle of 45°. How far does the Solution: θ= s r radius = 80 cm θ = 45° Convert 45° to π radians 45° x π rad = π rad 180 4 7

θ= s r π rad = s 4 80 s = π rad x 80 4 = 20πdoes the tip of the b. The minute hand of a clock is 5 cm long. How far hand travel in 35 min?Solution: Arc length formula = deg ree ( 2πr ) 180 360° in 60 min time or 360 = 6° 60 35 min ⇒ 35 x 6° = 120° L = 120 ( 2 ) ( 3.1416 ) ( 5 cm ) 360 = 18.33 cmTry this out! A. Determine whether each of the following points lie on the unit circle. 1.( 3 , 4 ) 55 8

2. ( 0.8, -0.6 )3. ( 2 2 , −1 ) 334. ( 2 , -1 )5. ( − 8 , −15 ) 17 17B. Express each radian measure in degrees. 1. 5π radians 3 2. 7π radians 10 3. −15π radians 12Express to the nearest seconds4. 3 radians5. -20.75 radiansC. Express each degree measure in radians1. 60° 2. 150° 3. 240° 6. -366°4. 780° 5. -300° 9. 108° 26′ 50″7. 22.5° 8. 75° 25′D. Solve the following. 9

swings a 1. The pendulum of a clock swings through an angle of 0.15 rad. If it distance of 30 cm, what is the length of the pendulum? 2. The minute hand of the clock is 10 cm long. How far does the tipof the hand move after 12 minutes? 3. An arc 15 cm long is measured on the circumference of a circleof radius 10 cm. Find an angle subtended at the center. Lesson 2 Arcs Rotations Along the Unit Circle An angle can be thought of as the amount of rotation generatedwhen a ray is rotated about its endpoints. The initial position of the ray is calledthe initial side of the angle and the position of the ray at the endpoint is calledterminal side. A clockwise rotation generates a negative angle while acounterclockwise rotation generates a positive angle. Positive angle Negative angleExample1: 2. 9π radians 4 Illustrate 1. 5π radians 2 4. −13π radians 4 3. 3π radians 10

5. 30° 6. -90° 7. -500° 8. 270°The positive side of the x-axis is the initial side1. 2.3. 4. 5. 6.-90° 30° 11

7. 8.-500° 270°Example 2: How many degrees is the angle formed when rotating ray makes a. 3 complete counterclockwise turns? b. 2 5 complete clockwise turns? 6 Solutions: a. 3 ( 360 )° = 1080° b. 2 5 ( -360° ) = -1020° 6Try this out!A. Draw an arc whose length is:1. 4π units 5. 7π units 12 12

2. 15π units 6. 3π units 103. -8π units 7. −15π units 24. − 3π units 8. π units 2 8B. Draw the following angle measures.1. 115° 2. -250°3. -620° 4. 390°C. A unit circle is divided into 12 congruent arcs._____________ 1. What is the length of its arcs? ______________2. What is the measure of the central angle determined bythis arc length?______________3. Wht is the central angle determined by an arc of length π ? 3_______________4. What is the measure of an angle determined by an arc oflength π ? 4_______________5. What is the length of the 6 arcs? 10arcs Lesson 3 Angles An angle whose vertex lies at the origin of the rectangular coordinatesystem and whose initial side is positive along the positive x-axis is said to be instandard position. 13

TerminalsideVertex initial sideExample 3.1 HDetermine whether each angle is in standard position.a. b. Tc. d.ES 14

Solution a. ∠T is an angle in standard position. b. ∠H is an angle in standard position. c. ∠E is not in standard position because the vertex is not at the point oforigin. d. ∠S is in standard position.Quadrantal Angles A quadrantal angle is an angle in standard position and whose terminalside lies on the x-axis or y-axis.Example 3.2 Determine whether each angle is a quadrantal angle. Give the measure ofeach angle. FG ∠F s not a quadrantal angle; it is not in standard position. ∠G is a quadrantal angle.Coterminal Angles Coterminal angles are angles having the same initial side and the sameterminal side. 15

BCExample 3.3 Determine the measure of the smallest positive angle coterminal with thegiven angle. a. 65° b. 128° c. -213° d. 654°Solution Angles coterminal with a given angle θ may be derived using theformula θ + 360n for all integers n.a. 65° + 360° =425° b. 128° + 360° = 488°c. -213° + 360° = 147° d. 654° + 360° = 1014°Example 3.4Find the coterminal angle less than 360° of each of the following. a. 750° b. 380° 16

c. 660° d. 820°a. 755° - 360°(20) = 35° b. 380° - 360° = 20°a. 660° - 360° = 300° b. 820° - 360°(2) =100°Reference Angles A reference angle is a positive acute angle formed between the x-axisand the terminal side of a given angle. 17

Example 3.5 In each of the following determine the quadrant in which the angle liesand determine the reference angle. a. 73° b. 135° c. 300° d. 920°Solution The reference angle can be derived using the formula 180°n ± θ. a. 73° 180°n ± θ. 73° terminates in Ql, hence 180°( 0 ) - θ = 73° θ = 73°, the reference angle is itselfb. 135° 135° terminates in Qll, hence 180°( 1 ) - θ = 135° θ = 180° - 135° θ = 45° is the reference anglea. b. 18

c. 300°300° terminates in Q1V , hence180°( 2 ) - θ = 300° θ = 360° - 300° θ = 60° is the reference angled. 920°First find the number of multiples of 180° in 920° 900° has 4 multiples of 180° and a remainder of 200° The terminal side of 200° is in Qlll. 180°( 1 ) - θ = 200° θ = 200° - 180° θ = 20° is the reference angleTry this outA. Determine the smallest positive angle coterminal with the given angle. 1. 57° 2. -250° 3. 94° 4. -175° 19

5. 116° 6. -349°7. 100° 30′ 8. 207° 55′9. 185° 08′ 10. 409° 28′ 45″B. Determine the quadrant in which the angle lies and find the reference angle.1. 84° 2. -140°3. 355° 4. -365°5. 290° 6. 480°7. -650° 8. 740°9. 330° 08′ 10. 204° 45′Let’s summarize The circle of radius one with center at origin is called the unit circle. Every point on the unit circle satisfies the equation x2 + y2 = 1. It intersect at the points ( 1,0 ), ( 0, 1 ), ( -1, 0 ) and ( 0, -1 ). 20

Converting Angle Measures To convert from degrees to radians, multiply the number of degrees by π . Then simplify. 180 To convert from radians to degree, multiply the number of radians by 180 . Then simplify. π θ An angle is the amount of rotation where one side is called the initial side and the other is the terminal side. θ An angle is in standard position if it is constructed in a rectangular coordinate system with vertex at the origin and the initial side on the positive side of the x-axis. θ Coterminal angle are angles having the same initial side and the same terminal side. θ Reference angle is an acute angle between the terminal side and the x-axis. θ To find the reference angle, write the angle in the form 180n ± θ where θ is the reference angle.What have you learnedAnswer the following correctly 1. If the equation of a unit circle is x2 + y2 = 1, does the point ( -5, 4 )lies on the unit circle circle? 2. Express 120° in radian measure. 3. What is the reference angle of -380°? 21

4. The coterminal angle less than 360° of 820° is __________.5. Convert − 7π rad to degree measure. 66. On a circle of radius 20cm, the arc intercepts a central angle of 1 rad. 5What is the arc length?7. At what quadrant is the terminal side of - 1080° located?8. How many degrees is the angle formed by a ray that makes 3 1 5 complete rotations counterclockwise?9. How many degrees is the angle formed by a ray that makes 2 2 3 complete rotations clockwise?10. A minute hand of a clock is 5 cm long. How far does the tip of the hand travel in 50 min? 22

Answer KeyHow much do you know1. b 2. b3. d 4. a5. a 6. c7. .52 8. 20.57 cm 10. 5π9. Qll 20Lesson 1A. 1. lies on the unit circle 2. lies on the unit circle 3. does not lie on the unit circle 4. does not lie on the unit circle 5. lie on the unit circleB. 1. 300° C 1. π rad 6. 61π rad 2. 392° 3 30 2. 5π rad 7. 5π rad 3 43. -225 3. 4π rad 8. 1.316 rad4. 171° 53′ 17″ 3 9. 1.893 rad5. -1188° 53′ 31″ 4. 13π rad 3 5. − 5π 3D. 1. 200 cm 2. 12.57 cm 3. 1.5 radLesson 2. 23

A. 1. 2. 3. 4.5. 6. 7. 8.B. 1. 2. 3. 4.C. 1. π 2. 30° 3. 60° 4. 45° 5. π , 5π 6 3Lesson 3 2. 110° 3. 454° A. 1. 417° 4. 185° 5. 470° 6. 110° 7. 460° 30′ 8. 567° 55′ 9. 545° 08′ 10. 769° 28′ 45″ B. 1. Ql , 84° 2. Qlll, 35° 3. Q lV, 5° 4. QlV, 5°5. Q lll, 70° 6. Q1, 60° 7. Qll, 10° 8. Q ll, 20° 9. Q1, 29°52′ 10. Q lll, 24° 45′What have you learned1. No 2. 2π rad 3. 20° 34. 100° 5. -21π rad 6. s = 4 cm 24

7. Q1 8. -960° 9. 1152°10. 26.18 cm 25

Module 1Circular Functions and Trigonometry What this module is about This module is about the unit circle. From this module you will learn thetrigonometric definition of an angle, angle measurement, converting degreemeasure to radian and vice versa. The lessons were presented in a very simpleway so it will be easy for you to understand and be able to solve problems alonewithout difficulty. Treat the lesson with fun and take time to go back if you thinkyou are at a loss. What you are expected to learnThis module is designed for you to: 1. define a unit circle, arc length, coterminal and reference angles. 2. convert degree measure to radian and radian to degree. 3. visualize rotations along the unit circle and relate these to angle measures. 4. illustrate angles in standard position, coterminal angles and reference angles.How much do you knowA. Write the letter of the correct answer.1. What is the circumference of a circle in terms of π ?a. π b. 2π c. 3π d. 4π2. An acute angle between the terminal side and the x-axis is called ______a. coterminal b. reference c. quadrantal d. right

3. 60 ° in radian measure is equal toa. π b. π c. π d. π 2 3 4 64. 2.5 rad express to the nearest seconds is equal toa.143° 14′ 24″ b. 143° 14′ 26″ c. 43° 14′ 26″ d. 43° 14′ 27″5. What is the measure of an angle subtended by an arc that is 7 cm if the radius of the circle is 5 cm?a. 1. 4 rad b. 1.5 rad c. 1.6 rad d. 1.7 rad6. Point M ( 24 , y) lie on the unit circle and M is in Q II. What is the value of 25y?a. 6 b. − 6 c. 7 d. − 7 25 25 25 257. What is the measure of the reference angle of a 315o angle?a. 45 o b. 15 o c. -45 o d. -15 o8. In which quadrant does the terminal side of 5π lie? 6a. I b. II c. III d. IV9. A unit circle is divided into 10 congruent arcs. What is the length of each arc?a. π b. π c. 2π d. 10π 10 5 5B. Solve:10. The minute hand of the clock is 12 cm long. Find the length of the arc traced by the minute hand as it moved from its position at 3:00 to 3:40. 2

What you will do Lesson 1 The Unit Circle A unit circle is defined as a circle whose radius is equal to one unit andwhose center is at the origin. Every point on the unit circle satisfies the equationx2 + y2 + 1.The figure below shows a circle with radius equal to 1 unit. If thecircumference of a circle is defined bythe formula c = 2πr and r = 1, thenc = 2π or 360° or 1 revolution. r =1If 2π = 360°, then π = 180° or ºone-half revolution.Example:1. Imagine the Quezon Memorial Circle as a venue for morning joggers. The maintainers have placed stopping points where they could relax. If each jogger starts at C BPoint A, the distance he would Atravel at each terminal pointis shown in table below. D Stopping B C D A Point Distance or π 3π Arclength 2 π 2π 2 This illustrates the circumference of the unit circle 2π when divided by 4:will give 2π = π , the measure of each arc. 42 3

Similarly, the measure of each arc of a unit circle divided into:a. 6 congruent arcs = 2π = π 63b. 8 congruent arcs = 2π = π 84c. 12 congruent arcs = π 6These measurements are called arclengths. Let’s go back to the unit circle which we divided into 4 congruent arcs.From A, the length of each arc in each terminal points is given as: π BB: + 2C: 2π = π CA 2 -D: 3π D 2A: 4π = 2π 2 This is true in a counterclockwise rotation. If the rotation goes clockwise,the arclengths would be negative. Thus, the arclengths of the terminal points in a clockwise direction wouldyield: π B = - 3πD=- 2 2C = -π A = -2πWe call these measurements as directed arclengths. 4

2. Suppose a point is allowed to move around the circle starting from point A,find the arclength of each terminal point.The unit circle is divided into C 8 congruent arcs. Therefore, DBeach arc measures π . EA 4 F H GA counterclockwise move that A clockwise move that terminates at: terminates at:Terminal pt. Arclength Terminal pt. Arclength π H B G 4 F C E D D E C F B G A H A B= π 4Units of Angle MeasuresThere are two unit of angle measure that are commonly used: 1. Degree measure 2. Radian measure. 5

A complete revolution is divided into 360 equal parts. Degree issubdivided to minutes and seconds. 1 rev = 360° 1° = 60′ ′ is the symbol for minutes 1′ = 60″ ″ is the symbol for seconds For all circles, the radian measure of the circumference is 2πradians. But the angle has a measure of 360°. hence, 2π rad = 360° π rad = 180° 1 rad = 180 or 57.296° π 1° = π rad or 0.017453 rad 180Converting Angle Measures To convert from degrees to radians, multiply thenumber of degrees by π . Then simplify. 180 To convert from radians to degree, multiply thenumber of radians by 180 . Then simplify. πExample 3:Convert the measure of the following angles from degrees to radians. a. 70° b. -225° c. 125° 15′ 45″ 6

Solution a. 70° = 70° x π = 7π rad 180 18b. -225° = -225° x π = − 5π rad 180 4c. 125° 15′ 45″Convert 15′ to seconds 900″ + 45″ = 945″ 60″ 15′ x 1′ = 900″Convert 945″ to degrees 945″ = 0.263° 125° + 0.263° = 125 .263° 3600″Since, 1° = π rad or 0.017453 rad 180Therefore, 125 .263° x 0.017453 rad = 2.186 radExample 4.Express each radian measure in degrees a. 2π b. 2.5 rad to the nearest seconds 4Solution a. 2π x 180 = 90° 4π 2.5 rad x 57.296° = 143.24° .24° x 60′ = 14.4′ 7

1° .4′ x 60″ = 24″ 1′∴ 2.5 rad = 143° 14′ 24″ Lesson 2.3 A radian is defined as the measure of an angle intercepting an arc whoselength is equal to the radius of the circle. An arc length is the distance betweentwo points along a circle expressed in linear units. arc length θ= sangle in radian = ________________ or rradius of the circle ))Example 5. Solve: a. A wheel of radius 80 cm rolls along the ground without slipping and rotates through an angle of 45°. How far does the wheel move?Solution: θ= s r radius = 80 cm θ = 45°Convert 45° to π radians 45° x π rad = π rad 180 4 θ= s r 8

π rad = s 4 80 s = π rad x 80 4 = 20πdoes the tip of the b. The minute hand of a clock is 5 cm long. How far hand travel in 35 min?Solution: Arc length formula = deg ree ( 2πr ) 180 360° in 60 min time or 360 = 6° 60 35 min ⇒ 35 x 6° = 120° L = 120 ( 2 ) ( 3.1416 ) ( 5 cm ) 360 = 18.33 cmTry this out! A. Determine whether each of the following points lie on the unit circle. 1.( 3 , 4 ) 55 2. ( 0.8, -0.6 ) 9

3. ( 2 2 , −1 ) 334. ( 2 , -1 )5. ( − 8 , −15 ) 17 17B. Express each radian measure in degrees. 1. 5π radians 3 2. 7π radians 10 3. −15π radians 12Express to the nearest seconds4. 3 radians5. -20.75 radiansC. Express each degree measure in radians1. 60° 2. 150° 3. 240° 6. -366°4. 780° 5. -300° 9. 108° 26′ 50″7. 22.5° 8. 75° 25′ D. Solve the following. 1. The pendulum of a clock swings through an angle of 0.15 rad. If itswings a 10

distance of 30 cm, what is the length of the pendulum? 2. The minute hand of the clock is 10 cm long. How far does the tipof the hand move after 12 minutes? 3. An arc 15 cm long is measured on the circumference of a circleof radius 10 cm. Find an angle subtended at the center. Lesson 2 Arcs Rotations Along the Unit Circle An angle can be thought of as the amount of rotation generatedwhen a ray is rotated about its endpoints. The initial position of the ray is calledthe initial side of the angle and the position of the ray at the endpoint is calledterminal side. A clockwise rotation generates a negative angle while acounterclockwise rotation generates a positive angle.Positive angle Negative angleExample1: 2. 9π radians Illustrate 1. 5π radians 4 2 3. 3π radians 4. −13π radians 4 5. 30° 6. -90° 11

7. -500° 8. 270°The positive side of the x-axis is the initial side1. 2.3. 4. 5. 6.-90° 30° 12

7. 8.-500° 270°Example 2: How many degrees is the angle formed when rotating ray makes a. 3 complete counterclockwise turns? b. 2 5 complete clockwise turns? 6 Solutions: a. 3 ( 360 )° = 1080° b. 2 5 ( -360° ) = -1020° 6Try this out!A. Draw an arc whose length is:1. 4π units 5. 7π units 12 13

2. 15π units 6. 3π units 103. -8π units 7. −15π units 24. − 3π units 8. π units 2 8B. Draw the following angle measures.1. 115° 2. -250°3. -620° 4. 390°C. A unit circle is divided into 12 congruent arcs._____________ 1. What is the length of its arcs? ______________2. What is the measure of the central angle determined bythis arc length?______________3. Wht is the central angle determined by an arc of length π ? 3_______________4. What is the measure of an angle determined by an arc oflength π ? 4_______________5. What is the length of the 6 arcs? 10arcs Lesson 3 Angles An angle whose vertex lies at the origin of the rectangular coordinatesystem and whose initial side is positive along the positive x-axis is said to be instandard position. 14

TerminalsideVertex initial sideExample 3.1 HDetermine whether each angle is in standard position.a. b. Tc. d.ES 15

Solution a. ∠T is an angle in standard position. b. ∠H is an angle in standard position. c. ∠E is not in standard position because the vertex is not at the point oforigin. d. ∠S is in standard position.Quadrantal Angles A quadrantal angle is an angle in standard position and whose terminalside lies on the x-axis or y-axis.Example 3.2 Determine whether each angle is a quadrantal angle. Give the measure ofeach angle. FG ∠F s not a quadrantal angle; it is not in standard position. ∠G is a quadrantal angle.Coterminal Angles Coterminal angles are angles having the same initial side and the sameterminal side. 16

BCExample 3.3 Determine the measure of the smallest positive angle coterminal with thegiven angle. a. 65° b. 128° c. -213° d. 654°Solution Angles coterminal with a given angle θ may be derived using theformula θ + 360n for all integers n.a. 65° + 360° =425° b. 128° + 360° = 488°c. -213° + 360° = 147° d. 654° + 360° = 1014°Example 3.4Find the coterminal angle less than 360° of each of the following. a. 750° b. 380° c. 660° d. 820° 17

a. 755° - 360°(20) = 35° b. 380° - 360° = 20°a. 660° - 360° = 300° b. 820° - 360°(2) =100°Reference Angles A reference angle is a positive acute angle formed between the x-axisand the terminal side of a given angle. 18

Example 3.5 In each of the following determine the quadrant in which the angle liesand determine the reference angle. a. 73° b. 135° c. 300° d. 920°Solution The reference angle can be derived using the formula 180°n ± θ. a. 73° 180°n ± θ. 73° terminates in Ql, hence 180°( 0 ) - θ = 73° θ = 73°, the reference angle is itselfb. 135° 135° terminates in Qll, hence 180°( 1 ) - θ = 135° θ = 180° - 135° θ = 45° is the reference anglea. b. 19

c. 300°300° terminates in Q1V , hence180°( 2 ) - θ = 300° θ = 360° - 300° θ = 60° is the reference angled. 920°First find the number of multiples of 180° in 920° 900° has 4 multiples of 180° and a remainder of 200° The terminal side of 200° is in Qlll. 180°( 1 ) - θ = 200° θ = 200° - 180° θ = 20° is the reference angleTry this outA. Determine the smallest positive angle coterminal with the given angle. 1. 57° 2. -250° 3. 94° 4. -175° 20

5. 116° 6. -349°7. 100° 30′ 8. 207° 55′9. 185° 08′ 10. 409° 28′ 45″B. Determine the quadrant in which the angle lies and find the reference angle.1. 84° 2. -140°3. 355° 4. -365°5. 290° 6. 480°7. -650° 8. 740°9. 330° 08′ 10. 204° 45′Let’s summarize The circle of radius one with center at origin is called the unit circle. Every point on the unit circle satisfies the equation x2 + y2 = 1. It intersect at the points ( 1,0 ), ( 0, 1 ), ( -1, 0 ) and ( 0, -1 ). 21

Converting Angle Measures To convert from degrees to radians, multiply the number of degrees by π . Then simplify. 180 To convert from radians to degree, multiply the number of radians by 180 . Then simplify. π θ An angle is the amount of rotation where one side is called the initial side and the other is the terminal side. θ An angle is in standard position if it is constructed in a rectangular coordinate system with vertex at the origin and the initial side on the positive side of the x-axis. θ Coterminal angle are angles having the same initial side and the same terminal side. θ Reference angle is an acute angle between the terminal side and the x-axis. θ To find the reference angle, write the angle in the form 180n ± θ where θ is the reference angle.What have you learnedAnswer the following correctly 1. If the equation of a unit circle is x2 + y2 = 1, does the point ( -5, 4 )lies on the unit circle circle? 2. Express 120° in radian measure. 3. What is the reference angle of -380°? 22

4. The coterminal angle less than 360° of 820° is __________.5. Convert − 7π rad to degree measure. 66. On a circle of radius 20cm, the arc intercepts a central angle of 1 rad. 5What is the arc length?7. At what quadrant is the terminal side of - 1080° located?8. How many degrees is the angle formed by a ray that makes 3 1 5 complete rotations counterclockwise?9. How many degrees is the angle formed by a ray that makes 2 2 3 complete rotations clockwise?10. A minute hand of a clock is 5 cm long. How far does the tip of the hand travel in 50 min? 23

Answer KeyHow much do you know1. b 2. b3. d 4. a5. a 6. c7. .52 8. 20.57 cm 10. 5π9. Qll 20Lesson 1A. 1. lies on the unit circle 2. lies on the unit circle 3. does not lie on the unit circle 4. does not lie on the unit circle 5. lie on the unit circleB. 1. 300° C 1. π rad 6. 61π rad 2. 392° 3 30 2. 5π rad 7. 5π rad 3 43. -225 3. 4π rad 8. 1.316 rad4. 171° 53′ 17″ 3 9. 1.893 rad5. -1188° 53′ 31″ 4. 13π rad 4. 3 8. 5. − 5π 3D. 1. 200 cm 2. 12.57 cm 3. 1.5 radLesson 2.A. 1. 2. 3.5. 7. 6. 24