Name: Pre-Calculus Ntes: Chapter Graphs f Trignmetric Functins Sectin 1 Angles and Radian Measure Angles can be measured in bth degrees and radians. Radian measure is based n the circumference f a unit circle (a circle with a radius f 1). Since the circumference f the unit circle is, 0 = radians. It is ften helpful t cnvert between degrees and radians. Example 1 a. Change 115 t exact (leave in the answer) radian measure. 115 180 b. Change 7 t degrees. 8 7 180 157.5 8 c. Change 15 t exact radian measure. d. Change t degrees. Example Use the unit circle (n calculatr) t evaluate. a. sin b. cs y value n unit circle c. cs 0 d. sin 5 e. tan f. 5 cs g. 1 sin cnvert t degrees 90 same as sine at 0 ½ h. 5 tan 1
Bth degree and radian measure can be used t calculate arc length and area f a sectr. Degrees Radians Arc Length x s r 0 s r Area f a Sectr x A r 1 A r 0 Example a. Given a central angle f 15, find the length f its intercepted arc in a circle f radius 7 centimeters. Rund t the nearest tenth. Angle is in degrees and I want arc length, s use x s r 0 15 s 7 15. cm 0 b. A pendulum with length f 1. meters swings thrugh an angle f 0. Hw far des the bb at the end f the pendulum travel as it ges frm left t right? Example a. Find the area f a sectr if the central angle measures and the radius f the circle 7 is 11 centimeters. Rund t the nearest tenth. Angle is in radians and I want area, s use 1 A r A 1 11 7 81.5cm b. A sectr has area f 15 square inches and central angle f 0. radians. Find the radius f the circle. Find the arc length f the sectr.
Sectin Graphing Sine and Csine Functins Peridic Functin Perid Example 1 Determine if each functin is peridic. If s, state the perid. a. b. c. d. Graphing the Csine Functin: y cs 0 5 7 5 5 7 11 cs 1 Dmain: Maximum: y-intercept Range: Minimum: x-intercept(s)
Graphing the Sine Functin: y sin 0 5 7 5 5 7 11 sin Dmain: Maximum: y-intercept Range: Minimum: x-intercept(s) Graphing the Tangent Functin: y tan 0 5 7 5 5 7 11 tan Dmain: Maximum: y-intercept Range: Minimum: x-intercept(s)
Example Find sin by referring t the graph f the sine functin. Example Find the value f fr which sin 1 is true. Example Graph y sin x fr x. Example 5 The graph at the right shws the average mnthly precipitatin (in inches) fr Seattle, Washingtn, and San Francisc, Califrnia, with January represented as 1. Mdel fr Seattle s precipitatin: Mdel fr San Francisc s precipitatin: y.5.55cs t 0. 18 y..cs t 0. 1 a. What is the average precipitatin fr each city fr mnth 1? b. Which city has the greater fluctuatin in precipitatin? Explain. 5
Sectin Amplitude and Perid f Sine and Csine Functins Sketch the fllwing functins n the axes belw. Set the windw f yur graphing calculatr t: xmin =, xmax =, xscl =, ymin = -, ymax =, yscl = 1. After graphing each equatin, fill in the table belw. y sin x y 5sin x 1 y sin x Perid Minimum Maximum Range y-intercept x-intercept(s) Amplitude
Sketch the fllwing functins n the axes belw. Set the windw f yur graphing calculatr t: xmin =, xmax =, xscl =, ymin = -, ymax =, yscl = 1. After graphing each equatin, fill in the table belw. y cs x y cs x y cs x Perid Minimum Maximum Range y-intercept x-intercept(s) Perid f Sine and Csine Functins Frequency Determine the amplitude and perid f each equatin. Amplitude Perid 1 a.) y sinx x b.) y cs x c.) y 5sin 1 d.) y csx 7
Example 1 Graph each functin. a. y cs b. y sin c. y 5cs Example Write an equatin f the sine functin with amplitude and perid. 8
Example A pendulum swings a ttal distance f 0.0 meter. The center pint is zer. It cmpletes a cycle every secnds. a. Assuming that the pendulum is at the center pint and heading right at t = 0, find an equatin fr the mtin f the pendulum b. Determine the distance frm a center at 1 secnd, 1.5 secnds, 1.75 secnds, and secnds. Example The Sears Building in Chicag sways back and frth at a vibratin frequency f abut 0.1 Hz. On average, it sways inches frm true center. Write an equatin f the sine functin that represents this behavir. Sectin 5 Translatins f Sine and Csine Functins A hrizntal translatin r shift f a trignmetric functin is called a phase shift. y A y Asin : Given the general frm, cs k c h r k c h Amplitude Perid Phase Shift Vertical Shift Examples: Graph each equatin. 1. y cs Amplitude Perid Phase Shift Vertical Shift 9
. y sin Amplitude Perid Phase Shift Vertical Shift. y sin Amplitude Perid Phase Shift Vertical Shift. y cs 1 Amplitude Perid Phase Shift Vertical Shift 10
5. y cs Amplitude Perid Phase Shift Vertical Shift. y cs Amplitude Perid Phase Shift Vertical Shift Examples: Use the given infrmatin t write an equatin. 7. Write an equatin f a csine functin with amplitude 5, perid, phase shift, and 8 vertical shift -. 8. Write an equatin f a sine functin with amplitude 7, perid, phase shift, and vertical shift 7. 11
Sectin Mdeling Real-Wrld Data with Sinusidal Functins Example1 An average seated adult breathes in and ut every secnds. The average minimum amunt f air in the lungs is 0.08 liter, and the average maximum amunt f air in the lungs is 0.8 liter. Suppse the lungs have a minimum amunt f air at t = 0, where t is time in secnds. a. Write a functin that mdels the amunt f air in the lungs. b. Determine the amunt f air in the lungs at 5.5 secnds. Example The tide in a castal city peaks every 11. hurs. The tide ranges frm.9 meters t. meters. Suppse that the lw tide is at t = 0, where t is time in hurs. a. Write a functin that mdels the height f the tide. b. Determine the height f the tide at. hurs. Example The average mnthly temperatures fr the city f Seattle, Washingtn, are given belw. Write a sinusidal functin that mdels the mnthly temperatures, using t = 1 t represent January. Jan. Feb. Mar. Apr. May June July Aug. Sept. Oct. Nv. Dec. 1 7 50 5 1 5 1 5 1
Sectin 7 Graphing Other Trignmetric Functins Graphing the Ctangent Functin: y ct 0 5 7 5 5 7 11 tan Dmain: Maximum: y-intercept Range: Minimum: x-intercept(s) Asmpttes: Perid: Graphing the Csecant Functin: y csc 0 5 7 5 5 7 11 tan Dmain: Maximum: y-intercept Range: Minimum: x-intercept(s) Asympttes: Perid: y = 1 y = -1 1
Graphing the Secant Functin: y sec 0 5 7 5 5 7 11 tan Dmain: Maximum: y-intercept Range: Minimum: x-intercept(s) Asympttes: Perid: y = 1 y = -1 Example 1 Find the values f fr which each equatin is true. a. csc 1 b. sec 1 c. tan 1 Example Find each value by referring t the graph. a. 7 ct 5 tan c. csc b. 8 1
Example Graph each equatin. y csc y sec 1 Example Write an equatin fr the given functin. a. Secant with perid, phase shift, and vertical shift -. b. Csecant with perid, phase shift, and vertical shift. 15
Sectin 8 Trignmetric Inverses and Their Graphs 1
Because we want t fcus n a part f the inverse that is a functin, we determine a restricted dmain fr wrking with inverses f sine, csine, and tangent. Examples Find each value. 1 1. Arc sin. 1 Cs. Sin 1 0 1. sin 1 1 Arc 5. Sin cs. Tan sin 7. cs 1 1 Arc tan Arc sin 8. sintan 1 Sin 1 9. Determine whether Sin -1 (sin x) = x is true r false fr all values f x. If false, give a cunterexample. 17