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Name: Group Members: Exploration 4-2a: Properties of Trigonometric Functions Date: Objective: Use the properties of trigonometric functions to transform an expression to another given form. 1. Write the four trigonometric functions tan x, cot x, sec x, and csc x in terms of sin x and cos x. 6. Divide both sides of the equation in Problem 5 by cos2 x. Simplify the result to get a Pythagorean property relating sec x and tan x. 2. The property sec x = cos1 x is called a reciprocal property. Why do you think this is the property’s name? sin x 3. What is the name of the property tan x = cos x? 7. Derive a Pythagorean property relating csc x and cot x. 4. Evaluate cos2 0.6 C sin2 0.6. Sketch a 0.6-radian angle in standard position, and use the drawing to explain the significance of your answer. 8. Derive another quotient property expressing tan x in terms of sec x and csc x. 5. What is the name of the property cos2 x + sin2 x = 1? (Over) Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Exploration Masters / 67 Name: Group Members: Exploration 4-2a: Properties of Trigonometric Functions continued 9. Transform the expression csc x tan x to sec x. Start by writing the given expression. Then substitute using appropriate properties and simplify. 10. The expression csc x tan x in Problem 9 involves two functions. The answer, sec x, involves only one function. What could be your thought process in deciding how to start this problem? 68 / Exploration Masters Date: 11. Transform csc x tan x cos x to 1. 12. What did you learn as a result of doing this Exploration that you did not know before? Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Name: Group Members: Exploration 4-3a: Transforming an Expression Date: Objective: Use the trigonometric properties to transform an expression to another given form. 1. Write the three reciprocal properties. 3. Write the three Pythagorean properties. 2. Write the two quotient properties. 4. Transform sec x D cos x to sin x tan x. Beside each step, write the technique you used from your “list of things to try.” 5. What did you learn as a result of doing this Exploration that you did not know before? Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Exploration Masters / 69 Name: Group Members: Exploration 4-3b: Trigonometric Transformations Date: Objective: Use the properties of functions of one argument to transform trigonometric expressions. 1. Without consulting text, notes, or other students, write 4. Transform csc P cos2 P C sin P to csc P. (Try factoring out the csc P you want, first.) • The three reciprocal properties: • The two quotient properties: • The three Pythagorean properties: 5. On your grapher, plot y1 = tan2 x and y2 = sec2 x. Use radian mode and a window with an x-range of [0, π] and a y-range of [0, 10]. By appropriate tracing, find out how the two graphs are related to each other. How does this relationship correspond to the trigonometric properties? Check your graph with your instructor. 2. Transform (1 C cos A)(1 D cos A) to sin2 A. 6. What did you learn as a result of doing this Exploration that you did not know before? 3. Transform cot B C tan B to csc B sec B. 70 / Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Name: Group Members: Exploration 4-3c: Trigonometric Identities Date: Objective: Use the trigonometric properties to transform an expression to another given form. No books or notes! csc x 5. Quick: Why does cot x = sec x? 1. Write the three reciprocal properties. 2. Write the two quotient properties. 6. Prove that sec A sin A sin A D cos A = cot A is an identity. 3. Write the three Pythagorean properties. 7. Prove: 4. Prove the alternate form of the quotient property, tan x = sec x csc x Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press 1 1 D cos B + 1 + 1cos B = 2 csc2 B 8. What did you learn as a result of doing this Exploration that you did not know before? Exploration Masters / 71 Name: Group Members: Exploration 4-4a: Arccosine, Arcsine, and Arctangent Date: Objective: Find values of arccosine, arcsine, and arctangent by calculator. 1. Find the degree measure of arccos 0.4 that equals cosD1 0.4. Sketch the angle in a uv-coordinate system (draw the arc and the arrow). Show the reference triangle and label two of its sides. 6. Write the general solution for arcsin 0.3. v u 7. Find sinD1 (D0.8) and a value of arcsin (D0.8) terminating in a different quadrant. Sketch the angles and label each reference triangle. v u 2. On the same axes, sketch another angle arccos 0.4 in a different quadrant. Sketch and label the reference triangle. Find the degree measure of the angle. 8. Find tanD1 5 and a value of arctan 5 terminating in a different quadrant. Sketch the angles and label each reference triangle. v 3. Write the general solution for arccos 0.4. u 9. Write the general solution for arctan 5. 4. Find the value of arcsin 0.3 that equals sinD1 0.3. Sketch the angle. Label two appropriate sides of the reference triangle. v 10. Find two values of arctan (D0.6) terminating in two different quadrants. Sketch the angles and label each reference triangle. u v u 5. Sketch another angle arcsin 0.3 in a different quadrant. Sketch and label the reference triangle. Find the measure of the angle. 11. What did you learn as a result of doing this Exploration that you did not know before? 72 / Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Name: Group Members: Exploration 4-4b: Trigonometric Equations Date: Objective: Solve equations in which trigonometric functions appear. 1. Solve: 2 cos (θ D 17−) = 1, θ E [0−, 720−] 4. The figure shows the graphs of y1 = D1 D 5 sin θ y2 = 2 cos2 θ 2. Solve: tan2 θ D 2 tan θ D 3 = 0, θ E [D360−, 360−] Show on the graph that all of your answers in Problem 3 are correct. y 5 180° 90° 270° 450° 630° θ 720° 5 5. Explain how what you have been studying about transformation of trigonometric expressions allows you to solve trigonometric equations. 3. Solve: D1 D 5 sin θ = 2 cos2 θ, θ E [D180−, 720−] Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Exploration Masters / 73 Name: Group Members: Exploration 4-5a: Parametric Function Pendulum Problem Date: Objective: Given a pendulum moving in both the x- and y-directions, predict its position, (x, y), at any given time, t. A pendulum hangs from the ceiling. The point on the floor above which the pendulum bob is situated when it is at rest is the origin, (0, 0), of a Cartesian coordinate system. The x- and y-axes run parallel to the walls of the room and the x-y plane is horizontal. 5. Using your equations in Problem 4, make a table of values of x and y for various values of t from 0 through 1 complete orbit of the pendulum. t x y 1. Give the pendulum bob a displacement of 30 cm in the positive x-direction. Let it go and time its period. Sketch the graph of x as a function of time, t seconds, since it was released. 6. Plot the points (x, y) and connect them. 2. Starting with the pendulum bob at (0, 0), give it a push in the y-direction just hard enough to make it swing with an amplitude of 20 cm. Does the pendulum have the same period this time? Sketch the graph of y as a function of t. y 25 x 40 3. Starting with the pendulum bob at (30, 0), give it a push in the y-direction just hard enough to make it have an amplitude of 20 cm in the y-direction, as in Problem 2. Sketch. Describe the path followed by the pendulum bob. 7. What geometrical figure describes the graph in Problem 6? 8. What special name is given to a variable, such as t in this problem, upon which two or more other variables depend? 9. What special name is given to functions in which two or more variables depend on the same independent variable? 4. Assume that the graphs in Problems 1 and 2 are sinusoids. Write particular equations for x and y as functions of t. xH 10. Put your grapher in parametric mode. Then enter the equations for x and y from Problem 4. Pick suitable ranges for x, y, and t. Have the calculator plot the graph. (Use ZOOM SQUARE to get equal scales on the axes.) Does the graph look like the graph you plotted in Problem 6? 11. Just for fun, see if you can transform the two equations from Problem 4 into one equation in only x and y by eliminating the parameter t. yH 74 / Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Name: Group Members: Exploration 4-5b: Parametric Equations for Ellipses Date: Objective: Given a figure involving ellipses, write parametric equations and draw the graph on your grapher. 4. Frustum of a cone (“truncated” cone): y 6 y 6 x 10 x 10 1. The figure shows an ellipse with center at (5, 3), x-radius H 4, and y-radius H 2. Write what you think the parametric equations of this ellipse are. xH Top: x1 H y1 H yH 2. Plot these parametric equations on your grapher. Use a window with equal scales on both axes. Does the graph agree with the figure? For Problems 3–5, write parametric equations for each ellipse shown. Plot the ellipses using degree mode with t E [0−, 360] and a t-step of 10−. Use appropriate Boolean variables and dot style for the parts of the ellipses that are hidden. Then use the DRAW command to draw line segments and dots in appropriate places. Bottom, solid (use a Boolean variable) x2 H y2 H Bottom, dashed (use a Boolean variable) x3 H y3 H 5. Cone oriented the other way: y 3. Cone: 6 y 6 x 10 x 10 Solid (use a Boolean variable) xH x1 H yH y1 H Dashed (use a Boolean variable) x2 H y2 H 6. What did you learn as a result of doing this Exploration that you did not know before? Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Exploration Masters / 75 Name: Group Members: Exploration 4-6a: Graphs of Inverse Trigonometric Relations Date: Objective: Investigate the graphs of inverse circular functions and relations by plotting them on your grapher. 4. Come back to function mode. Plot the graph of the function y H sinD1 x. Because the calculator gives only the principal value of the inverse sine, the graph is called the principal branch. Darken the part of the given graph that corresponds to the principal branch. This graph shows the inverse circular relation y = arcsin x In this Exploration, you will learn how to plot this graph on your grapher. y 5. Return to the parametric mode and enter the parent sine function graph. Enter x2T = t y2T = sin t 1 x 1 Keep the x1T, y1T graph active. Sketch the resulting graph on the given figure. 6. What relationship do the sine and inverse sine graphs have to each other? How do they relate to the line y H x? 1. Recall that y H arcsin x means that x H sin y. Calculate x for y H 4. 2. Write the general solution for y H arcsin x. Use it to find the four values of arcsin 0.4 that are on the portion of the graph shown. Round to one decimal place. Plot points on the graph corresponding to the four answers. 7. Plot the graphs of y H cos x and y H arccos x on your grapher. Use the same window as in the previous problems. Sketch the result here, along with the line y H x. y 1 x 1 3. Reproduce the given graph on your grapher. A clever way to do this is to use parametric mode. Enter the equation as x1T = sin t y1T = t Use a t-range of D9 to 9. Use [D9, 9] for the x-range and equal scales for the two axes. Check the graph with your instructor. 76 / Exploration Masters 8. What did you learn as a result of doing this Exploration that you did not know before? Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Name: Group Members: Exploration 4-6b: Principal Branches of Inverse Trigonometric Relations Date: Objective: Figure out the principal branches of each of the six inverse circular functions. 1. On your grapher, plot the inverse circular function y H sinD1 x. Use equal scales on the two axes. On the graph of y H arcsin x shown here, darken the principal branch of the inverse sine relation on the part of the graph that is the inverse sine function. Give the range of the principal branch. 4. Use the technique in Problems 1 and 2 to find the range of y H tanD1 x. Darken the principal branch on this graph of y H arctan x. Range: y Range: y 1 x 1 1 x 1 2. Use the technique in Problem 1 to find the range of y H cosD1 x. Darken the principal branch on the graph of y H arccos x, shown next. 5. The range of y H sinD1 x is the closed interval CD π2 , π2 D. Explain why the range of y H tanD1 x cannot include the endpoints. Range: y 1 x 1 3. Why can’t the range of the inverse cosine function (y H cosD1 x) be the same as the range of the inverse sine function (y H sinD1 x)? 6. If the range of the inverse tangent function (y H tanD1 x) were [0, π] (excluding π2 ), like the range of y H cosD1 x, then y H tanD1 x would still be a function. What disadvantage would there be to defining the range of y H tanD1 x this way? (Over) Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Exploration Masters / 77 Name: Group Members: Exploration 4-6b: Principal Branches of Inverse Trigonometric Relations continued Date: 7. Duplicate the previous graph of y H arctan x on your grapher. Give the parametric equations you used. Check your graph with your instructor. 10. Look in the text to find out if the principal branch you chose in Problem 9 is the commonly accepted one. 8. The graph here shows y H arccot x. How can you define the range of the function y H cotD1 x in such a way that the function is continuous? Darken this principal branch of y H arccot x. 11. This next graph shows y H arccsc x. Shade what you think the principal branch is. Write the range of the function you shaded. Range: Range: y y 1 1 x x 1 1 12. Does your answer to Problem 11 agree with the range listed in the text? 9. This next graph shows y H arcsec x. There is no way to restrict the range to make a continuous function y H secD1 x and still use all of the domain. Darken what you think would be the best choice for the principal branch. Write the range. Range: y 13. Look up in the text the five criteria for picking the ranges of the principal branches of the six inverse circular functions. Write the criteria here. 1 x 1 14. What did you learn as a result of doing this Exploration that you did not know before? 78 / Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Name: Group Members: Exploration 4-6c: Inverse Trigonometric Relation Values Date: Objective: Calculate multiple values of inverse circular relations and confirm them graphically. y y y 10 10 5 5 10 5 x 5 x 2 2 5 x 2 2 5 5 5 1. Evaluate arcsin 0.8 for all values that show on the graph of y H arcsin x at the top of this column. Round final answers to the nearest 0.1 unit. Mark the solutions on the graph. 2. Evaluate arccos (D0.3) for all values that show on the graph of y H arccos x at column top. Round final answers to the nearest 0.1 unit. Mark the solutions on the graph. Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press 3. Evaluate arctan 4 for all values that show on the graph at the top of this column, of y H arctan x. Round final answers to the nearest 0.1 unit. Mark the solutions on the graph. 4. Darken the principal branch of each graph. Circle the point your calculator gives for sinD1 0.8, cosD1 (D0.3), and tanD1 4 on the respective graphs, showing in each case that the value is on the principal branch. 5. What did you learn as a result of doing this Exploration that you did not know before? Exploration Masters / 79 4. 3 radians = 3 • 5. 240− = 240− • 4. cos2 0.6 + sin2 0.6 = 1 180− = 171.8873… π v 4π π = radians 180− 3 sin 0.6 1 7 6. radians 10 u cos 0.6 7. y 5 5. Pythagorean property 3 6. cos2 x + sin2 x = 1 1 1 • (cos2 x + sin2 x) = 2 cos2 x cos x cos2 x sin2 x 1 + = cos2 x cos2 x cos2 x sin x 2 1 2 = 1+ cos x cos x 1 + tan2 x = sec2 x x 5 8. y = 5 D 2 cos 10 15 20 25 π π x or y = 5 + 2 cos (x D 4) 4 4 9. y(27) = 6.4142… m M 6.4 m 10. x = 14.9872… m M 15.0 m 11. 7 feet π (x D 1) = 7 ⇒ x = 1 a.m. 5.8 Period = 2 • 5.8 = 11.6 hours 3 + 4 cos π (x D 1) = D1 ⇒ x = 6.8 = 6:48 a.m. 5.8 1 foot deep 12. 3 + 4 cos 13. 4:00 p.m. is x = 16; y(16) = 1.9298… ft M 1.9 ft, which agrees with the graph. 14. Because this happens at the end of the second complete π cycle, it is where 5.8 (x D 1) = 4π ⇒ x = 24.2 hr = 12:12 a.m. on January 2. 15. 5.465… hr ≤ x ≤ 8.1343… hr or approximately 5:28 a.m. ≤ x ≤ 8:08 a.m. 5.8 D3 π (x D 1) = 0 ⇒ x = 1 + cosD1 5.8 π 4 = 5.4656… hr M 5:28 a.m. 7. cos2 x + sin2 x = 1 1 1 • (cos2 x + sin2 x) = 2 sin2 x sin x cos2 x sin2 x 1 + = sin2 x sin2 x sin2 x cos x 2 1 2 b +1= sin x sin x cot2 x + 1 = csc2 x 8. tan x = sin x 1/cos x sec x = = cos x 1/sin x csc x 9. csc x • tan x = 1 sin x 1 • = = sec x sin x cos x cos x 10. Answers will vary. 1 sin x • • cos x sin x cos x 1 = • cos x = 1 cos x 11. csc x • tan x • cos x = 16. 3 + 4 cos 17. Answers will vary. Chapter 4 • Trigonometric Function Properties, Identities, and Parametric Functions Exploration 4-2a sin x cos x cos x cot x = sin x 1 sec x = cos x 1 csc x = sin x 1. tan x = 12. Answers will vary. Exploration 4-3a 1 tan x 1 sec x = cos x 1 csc x = sin x 1. cot x = sin x sec x = cos x csc x cos x csc x cot x = = sin x sec x 2. tan x = 3. cos2 x + sin2 x = 1 1 + tan2 x = sec2 x cot2 x + 1 = csc2 x 2. One function is the reciprocal of the other. 3. Quotient property Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Solutions to the Explorations / 243 4. sec x D cos x ⇒ sin x tan x sec x D cos x 1 = D cos x cos x Exploration 4-3c 1. sec x = 1 1 1 , csc x = , cot x = cos x sin x tan x Make fractions to add for 1 term. cos2 x 1 D = cos x cos x 2. tan x = sin x cos x , cot x = cos x sin x Get common denominator. 1 D cos2 x = cos x 4. tan x = 3. cos2 x + sin2 x = 1, tan2 x + 1 = sec2 x, cot2 x + 1 = csc2 x sin x cos x 1/csc x = 1/sec x sec x = csc x Add to get 1 term. sin2 x = cos x When you see squares of functions, think Pythagorean. sin x = sin x • cos x If you see something you want in the answer, guard it with your life. sin x • tan x Familiar property 5. cot x = 6. 5. Answers will vary. Exploration 4-3b 1 1 1 , csc x = , cot x = cos x sin x tan x sin x cos x tan x = , cot x = cos x sin x cos2 x + sin2 x = 1, tan2 x + 1 = sec2 x, cot2 x + 1 = csc2 x 1. sec x = 2. (1 + cos A)(1 D cos A) = 1 D cos2 A = sin2 A cos B sin B + sin B cos B 2 cos B sin2 B = + sin B cos B sin B cos B cos2 B + sin2 B = sin B cos B 1 = sin B cos B = csc B sec B 3. cot B + tan B = 1 sin P ) csc P 2 2 = csc P (cos P + sin P ) = csc P 4. csc P cos2 P + sin P = csc P (cos2 p + 7. 1 1 csc x = = tan x sec x/csc x sec x sin2 A sec A sin A 1/cos A D = D sin A cos A sin A cos A sin A 1 sin2 A = D cos A sin A cos A sin A 1 D sin2 A = cos A sin A cos2 A = cos A sin A cos A = sin A = cot A 1 1 + 1 D cos B 1 + cos B 1 + cos B 1 D cos B = + (1 + cos B)(1 D cos B) (1 D cos B)(1 + cos B) 1 + cos B 1 D cos B = + 1 D cos2 B 1 D cos2 B 1 + cos B + 1 D cos B = 1 D cos2 B 2 = sin2 B = 2 csc2 B 8. Answers will vary. Exploration 4-4a 1. arccos 0.4 H 66.4218…− v 1 5. θ y u 0.4 5 y2 y1 x π 2 y2 appears to be 1 + y1. This is consistent with the property sec2 x = tan2 x + 1. 6. Answers will vary. 244 / Solutions to the Explorations Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press 2. arccos 0.4 H Y66.4218…− v 9. arctan 5 = 78.6900…− + n180− or 158.6900…− + n360− 10. arctan (D0.6) = D30.9637…− or 149.6362…− 1 v θ u θ 1 0.4 1 u θ θ 1 3. arccos 0.4 H J66.4218…− C 360n− Original and new θ have the same reference angle. 4. arcsin 0.3 H 17.4576…− 11. Answers will vary. v 0.3 1 θ 1. 2 cos (θ D 17−) = 1 ⇒ cos (θ D 17−) = u 5. arcsin 0.3 H 162.5423…− v 0.3 0.3 1 θ 1 θ Exploration 4-4b u 6. arcsin 0.3 H 17.4576…− C n360− or 162.5423…− C n360− 1 2 ⇒ θ D 17− = ±60 + n360− ⇒ θ = 53− + n360− or D77− + n360− = 53−, 283−, 413−, 643− 2. tan2 θ D 2 tan θ D 3 = (tan θ D 3)(tan θ + 1) = 0 ⇒ tan θ = 3 or tan θ = D 1 ⇒ θ = 71.5650…− + n180− or θ = D 45− + n180− θ = D288.4349…−, D225−, D108.4349…−, D45−, 71.5650…−, 135−, 251.5650…−, 315− 3. D1 D 5 sin θ = 2 cos2 θ = 2 D 2 sin2 θ ⇒ 2 sin2 θ D 5 sin θ D 3 = 0 ⇒ (2 sin θ + 1)(sin θ D 3) = 0 ⇒ sin θ = D 12 (Note: sin θ < 3 for all θ) ⇒ θ = D30− + n360− or θ = D150− + n360− θ = D150−, D30−, 210−, 330−, 570−, 690− 4. 7. sinD1 (D0.8) H D53.1301…− arcsin (D0.8) H 180− D (D53.1301…−) H 233.1301…− y 5 v θ 90° 0.6 0.8 0.6 θ θ 1 1 u 270° 180° 450° 630° 720° 5 0.8 5. Answers will vary. 8. tan D15 = 78.6900…− arctan 5 = 78.69000…− + 180− = 158.6900…− Exploration 4-5a 1. Example v x (cm) 1 30 θ u t (sec) p θ 1 Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press 30 Solutions to the Explorations / 245 10. 2. Example y y (cm) 30 10 t (sec) x 10 p 30 Period should be the same. 2π x 2π y t = , sin t= P 30 P 20 2π 2π cos2 t + sin2 t=1 P P 2 2 x y ⇒ + =1 30 20 11. cos 3. Path is an ellipse. 4. If the period is P seconds, then 2π x = 30 cos t P 2π t y = 20 sin P Exploration 4-5b 5. t x 0 30 P 8 21.2132… P 4 0 0 3P 8 D21.2132… P 2 D30 5P 8 D21.2132… 3P 4 1. x H 5 C 4 cos t y H 3 C 2 sin t y 2. Equations are correct. 14.1421… 3. x H 4 C 2 cos t, y H 5 C 0.4 sin t 20 4. Top: x H 4 C 2 cos t, y H 5 C 0.4 sin t 14.1421… 0 D14.1421… D20 0 7P 8 21.2132… P 30 D14.1421… 0 6. Bottom: (solid) x H 4 C 3 cos t/(t L 0 and t K π), y H 1 C 0.6 sin t/(t L 0 and t K π) Bottom: (dashed) x H 4 C 3 cos t/(t L π and t K 2π), y H 1 C 0.6 sin t/(t L π and t K 2π) 5. (solid) x H 1 C 0.4 cos t/(t L π2 and t K 3π 2 ), y H 3 C 2 sin t/(t L π2 and t K 3π 2 ) (dashed) x H 1 C 0.4 cos t/(t L D π2 and t K π2 ), y H 3 C 2 sin t/(t L D π2 and t K π2 ) 6. Answers will vary. y 2 Exploration 4-6a x 40 1. x H sin 4 H D0.7568… 2. arcsin x H sinD1 x C 2πn or (π D sinD1 x) C 2πn y H arcsin 0.4 H 0.4115… C 2πn or 2.7300… C 2πn y H D5.8716…, D3.5531…, 0.4115…, 2.7300… y ≈ D5.9, D3.6, 0.4, 2.7 y 7. The graph is an ellipse. 8. Parameter 9. Parametric function 246 / Solutions to the Explorations 1 x 1 Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press 4. Range: D π2 ≤ y ≤ π2 3. y y 1 x 1 1 x 1 4. y 1 x 5. tan π2 and tan D π2 are undefined. So the range of y H tanD1 x cannot include these numbers. ( ) 1 6. The graph would not be continuous. 7. x H tan 5; y H t 5. y 8. Range: 0 < y < π y 1 x 1 1 6. They are inverses of each other. Their graphs are reflections across the line y H x. x 1 7. y 1 9. Range: 0 ≤ y ≤ π, and y ≠ π2 x y 1 1 8. Answers will vary. x 1 Exploration 4-6b 1. Range: D π2 ≤ y ≤ π2 y 10. It is the commonly accepted branch. 1 11. Range: D π2 ≤ y ≤ π2 , and y ≠ 0 x y 1 2. Range: 0 ≤ y ≤ π 1 y x 1 1 x 1 12. Agrees D1 D1 3. If the range of cos were the same as that of sin , then cosD1 would not be a function. Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press Solutions to the Explorations / 247 13. 1. Must be a function 2. Must use entire domain 3. arctan 4 H 1.3258… C πn ≈ D5.0, D1.8, 1.3, 4.5, 7.6, 10.7 y 3. Should be continuous 4. Should be centrally located 10 5. If there is a choice, choose the positive branch. 14. Answers will vary. Exploration 4-6c 5 1. arcsin 0.8 H 0.9272… C 2πn or 2.2142… C 2πn ≈ D5.4, D4.1, 0.9, 2.2, 7.2, 8.5 y x 5 5 10 5 5 4. The values are on the principal branches. 5. Answers will vary. x 2 2 Chapter 5 • Properties of Combined Sinusoids Exploration 5-2a 1. y1 is the solid graph, y2 the dashed graph. 5 2. y 2. arccos (D0.3) H ±1.8754… C 2πn ≈ D44, D1.9, 1.9, 4.4, 8.2, 10.7 5 θ y 360° 10 3. A = 5 D = 53.1301…− 4. y4 = 5 cos (θ D 53.1301…−) The graph coincides. y 5 5 θ 360° x 2 2 5 5. A cos D = 3 A sin D = 4 A = √A2 = √A2(cos2 D + sin2 D) = √(A cos D)2 + (A sin D)2 = √32 + 42 =5 6. 3 = A cos D = 5 cos D ⇒ cos D = 4 = A sin D = 5 sin D ⇒ sin D = D = cosD1 35 = sinD1 45 = 53.1301…− 248 / Solutions to the Explorations Precalculus with Trigonometry: Instructor’s Resource Book, Volume 1 ©2003 Key Curriculum Press