A Sense of Déjà Vu Periodic Functions

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1 Lesson. Skills Practice Name Date A Sense of Déjà Vu Periodic Functions Vocabular Write the term that best comletes each statement.. The terminal ra of an angle in standard osition is the ra with its endoint at the origin that is not the.. The of a eriodic function is one half the absolute value of the difference between the maimum and minimum values of the function.. An angle is in when the verte is at the origin and one ra of the angle is on the -ais.. A is a function whose values reeat over regular intervals. 5. The of a eriodic function is a reference line whose equation is the average of the minimum and maimum values of the function. 6. The of a eriodic function is the length of the smallest interval over which the function reeats. 7. The measure of an angle in standard osition is the amount of rotation from the initial ra to the. Chater Skills Practice 75

2 Lesson. Skills Practice age Problem Set Sketch a grah for the function that is described in each eercise.. On an obstacle course, a erson must jum from a latform onto a giant addle wheel that has a diameter of 7 ards. The erson must then ride the addle wheel for at least one revolution before juming off the wheel and back onto the latform. Diana rides the wheel for three revolutions before juming off. The latform is nine ards off the ground. Sketch a grah of the model of the height of Diana above the ground with resect to the number of revolutions. Height of Person (ards) 5 5 Number of Revolutions. A frog clings to the edge of a addle of a wheel that is sinning behind a addle boat. The wheel has a diameter of 6 feet. The frog hos on the edge of the wheel just after it comes out of the water and manages to sta on for four revolutions before falling into the water. Sketch a grah of the model of the height of the frog above water with resect to the number of revolutions. 75 Chater Skills Practice

3 Lesson. Skills Practice age Name Date. A water ark has just oened the first ever underwater Ferris wheel. It is 5 feet in diameter. Each rider gets into a comletel waterroof car at ground level and rides around for an underwater adventure for five revolutions. Then, the eit the ride at ground level. Sketch a grah of the height of the rider in relationshi to ground level with resect to the number of revolutions.. A fl leaves its sot on the to of a bookshelf and lands a horizontal distance of.5 feet awa on the end of a blade of a ceiling fan. It rides around for four revolutions before fling back to its original sot on the bookshelf. The ceiling fan has a diameter of four feet. Sketch a grah of the model of the distance the fl is from the sot on the bookshelf with resect to the number of revolutions. Chater Skills Practice 755

4 Lesson. Skills Practice age 5. Franco s favorite ride at the fair is the -foot-diameter carousel. He hos on the black horse while his grandmother stands outside the gate at the nearest oint to him. Before the ride begins, Franco is feet from his grandmother. The carousel goes around 8 times during the ride and then stos in its original osition. Sketch a grah of the model of the distance Franco is from his grandmother with resect to the number of revolutions. 6. A bug hos on the ti of a 7-inch-long second hand of a mantel clock while it is ointing to the and rides the second hand around the clock. The bug stas on the second hand for 5 full minutes before juming off. Sketch a grah of the model of the height of the bug above the mantel with resect to the number of revolutions. Assume that the distance between the mantel and the 6 on the clock is inches. 756 Chater Skills Practice

5 Lesson. Skills Practice age 5 Name Date Determine whether each grah reresents a eriodic function over the interval shown. If so, identif the eriod Yes. The grah reresents a eriodic function with eriod Chater Skills Practice 757

6 Lesson. Skills Practice age Determine the midline and amlitude of each grah... The midline is 5 because () 5. The amlitude is because () Chater Skills Practice

7 Lesson. Skills Practice age 7 Name Date Chater Skills Practice 759

8 Lesson. Skills Practice age Chater Skills Practice

9 Lesson. Skills Practice Name Date Two Pi Radii Radian Measure Vocabular Comlete each sentence with the word(s) that makes it true.. A unit circle has a radius of.. A smbol used to identif the central angle measure of a circle in standard osition is.. There are in 6. Problem Set Calculate the arc length and the radian measure of each angle in a circle with the given dimensions.. 5 ; radius 5 units The radian measure is 6 radian. Arc length 5 ()? 6 5 (6) ( ) 5 units Radian measure = 7 ; radius 5 units Chater Skills Practice 76

10 Lesson. Skills Practice age. 5 5 ; radius 5 5 units. 5 ; radius 5 6 units ; radius 5 units ; diameter 5 units 76 Chater Skills Practice

11 Lesson. Skills Practice age Name Date ; diameter 5 units 8. = 5 ; diameter 5 5 units 9. u 5 6 ; diameter 5 8 units. u 5 ; diameter 5 units Chater Skills Practice 76

12 Lesson. Skills Practice age Estimate the degree measure of each central angle given in radians in a unit circle. Elain our reasoning.. radians The degree measure of radians is a little less than. Four radians is two-thirds of si radians, which is a little less than 6. Two-thirds of 6 is. So, four radians is a little less than.. radians. 5 radians. 6 radians 5. radian 76 Chater Skills Practice

13 Lesson. Skills Practice age 5 Name Date 6..5 radians 7..5 radians radians radians. 6.8 radians Chater Skills Practice 765

14 Lesson. Skills Practice age 6 Convert each radian measure to degrees. Round each answer to the nearest hundredth.. 5 radians Degree measure 5 5 radians? 8 radians radians. 6 radians. radians 5. radians 6..9 radians radians 8.. radians radians.. radians 766 Chater Skills Practice

15 Lesson. Skills Practice age 7 Name Date Convert each degree measure to radians. Write each answer as a simlified ratio in terms of.. Radian measure 5? radians radians Chater Skills Practice 767

16 Lesson. Skills Practice age Chater Skills Practice

17 Lesson. Skills Practice Name Date Triangulation The Sine and Cosine Functions Vocabular Write a definition for each term in our own words.. sine function. cosine function. trigonometric function. eriodicit identit Chater Skills Practice 769

18 Lesson. Skills Practice age Problem Set Use the unit circle to determine each value.. sin ( ). sin ( 5 sin ( ) 5 6 ). sin(). sin ( 5 ) 5. cos ( ) 6. cos ( ) 7. cos ( ) 8. cos ( 7 6 ) 77 Chater Skills Practice

19 Lesson. Skills Practice age Name Date The measure of an angle is given in radians. Determine the coordinates of the oint at which the terminal ra of each angle intersects the unit circle. 9. radians. radians (, ). radians. 6 radian. radians. 7 radians 5. 5 radians 6. radians 7. radians 8. 6 radians Chater Skills Practice 77

20 Lesson. Skills Practice age Evaluate the sine and cosine of the sulement of the given measure. Elain our reasoning sin ( cos ( 5 6 radian ) 6 radian ) 5 The sulement of 5 radians or 5 is or. This is equivalent to 6 radian Chater Skills Practice

21 Lesson. Skills Practice Name Date Pum U the Amlitude Transformations of Sine and Cosine Functions Vocabular Write the word(s) that best comletes each statement.. The of a eriodic function is the recirocal of the eriod and secifies the number of reetitions of the grah of a eriodic function er unit.. For eriodic functions, a horizontal translation is called a. Problem Set Determine the amlitude of each grah.. 5 sin(). 5 cos() amlitude 5 Chater Skills Practice 77

22 Lesson. Skills Practice age. 5 cos ( ). 5 sin() cos() 6. 5 sin() 77 Chater Skills Practice

23 Lesson. Skills Practice age Name Date Determine the eriod and frequenc of each grah cos() 8. 5 sin() The eriod is, or radians. The frequenc is sin ( ). 5 cos() Chater Skills Practice 775

24 Lesson. Skills Practice age. 5 sin(). 5 cos ( ) Describe the transformation erformed on the grah of the basic function f() to roduce the grah of g().. f() 5 sin(), g() 5 6 sin() The grah of f() is stretched verticall b a factor of 6.. f() 5 sin(), g() 5 sin ( 5 ) 5. f() 5 sin(), g() 5 sin() 6. f() 5 sin(), g() 5 sin( ) 776 Chater Skills Practice

25 Lesson. Skills Practice age 5 Name Date 7. f() 5 cos(), g() 5 cos(5) 8. f() 5 cos(), g() 5 6 cos() 9. f() 5 cos(), g() 5 cos ( ). f() 5 cos(), g() 5 cos() 7 Use what ou know about transformations to sketch the grah of each given function.. 5 cos(). 5 cos() Chater Skills Practice 777

26 Lesson. Skills Practice age 6. 5 sin ( ). 5 sin() 5. 5 sin( ) 6. 5 cos ( ) 778 Chater Skills Practice

27 Lesson.5 Skills Practice Name Date Farmer s Tan The Tangent Function Vocabular. Provide an eamle of a tangent function.. Elain how the tangent function is related to the sine and cosine functions. Problem Set Calculate the tangent of each angle given the cosine and sine of the angle.. sin(u) 5, cos(u) tan(u) 5 sin(u) cos(u) sin(u) 5 7, cos(u) sin(u) 5 8 7, cos(u) sin(u) 5, cos(u) 5 5 Chater Skills Practice 779

28 Lesson.5 Skills Practice age 5. sin(u) 5 cos(u) sin(u) 5, cos(u) sin(u) 5 5, cos(u) sin(u) 5, cos(u) Evaluate each tangent function b using the relationshi between the tangent function and the sine and cosine functions. 9. tan ( ) tan ( ) 5 5 undefined. tan(). tan ( 5 ). tan ( ) 78 Chater Skills Practice

29 Lesson.5 Skills Practice age Name Date. tan(). tan ( ) 5. tan ( ) 6. tan ( ) 7. tan ( 7 6 ) 8. tan ( 6 ) 9. tan ( ). tan ( 5 6 ) Chater Skills Practice 78

30 Lesson.5 Skills Practice age Comare the grah of each transformation to the grah of tan() shown below. Then, answer the question and elain how ou determined our answer.. Does the grah below reresent the function 5 tan() or 5 tan()? The grah reresents the function 5 tan(), because multiling the function b causes a vertical stretch. 78 Chater Skills Practice

31 Lesson.5 Skills Practice age 5 Name Date. Does the grah below reresent the function 5 tan() or 5 tan ( )?. Does the grah below reresent the function 5 tan() or 5 tan()? Chater Skills Practice 78

32 Lesson.5 Skills Practice age 6. Does the grah below reresent the function 5 tan ( ) or 5 tan ( )? 5. Does the grah below reresent the function 5 tan() or 5 tan()? 78 Chater Skills Practice

33 Lesson.5 Skills Practice age 7 Name Date 6. Does the grah below reresent the function 5 tan() or 5 tan()? Grah each transformation of the function tan() tan() 8. 5 tan ( ) Chater Skills Practice 785

34 Lesson.5 Skills Practice age tan ( ). 5 tan(). 5 tan(). 5 tan() 786 Chater Skills Practice

6.1 - Introduction to Periodic Functions

6.1 - Introduction to Periodic Functions Periodic Functions: Period, Midline, and Amplitude In general: A function f is periodic if its values repeat at regular intervals. Graphically, this means that