Trig Equations PS Sp016 NAME: SCORE: INSTRUCTIONS PLEASE RESPOND THOUGHTFULLY TO ALL OF THE PROMPTS IN THIS PACKET. TO COMPLETE THE TRIG EQUATIONS PROBLEM SET, YOU WILL NEED TO: 1. FILL IN THE KNOWLEDGE & COMPREHENSION COMPENDIUM.. WORK OUT APPLICATION & ANALYSIS PRACTICE PROBLEMS FROM EUREKA MATH LESSON 3 PROBLEM SET #1-6, LESSON 4 PROBLEM SET #1-7, LESSON 6 PROBLEM SET #1-3, LESSON 1 PROBLEM SET #1-3, ADDENDUM TO LESSON 1 PROBLEM SET #1-14. 3. WORK OUT SYNTHESIS & EVALUATION PRACTICE PROBLEMS #1-. FOR EXTRA PRACTICE IN PREPARATION FOR MAJOR ASSESSMENTS, PLEASE CONSIDER DOING ANY OF THE FOLLOWING: 1. KC READINESS ASSESSMENT: WRITING EACH KC COMPENDIUM PROMPT ON ONE SIDE OF A FLASH CARD AND YOUR RESPONSE ON THE OTHER SIDE. THEN, QUIZ YOURSELF NIGHTLY BEFORE GOING TO BED UNTIL YOU RECALL THEM ALL PERFECTLY.. AA READINESS ASSESSMENT: WORK OUT SOLUTIONS TO THE ODD- NUMBERED EXERCISES IN SECTION 5.3-5.5 OF YOUR PRECALCULUS TEXTBOOK, AND CHECK YOUR ANSWERS IN THE APPENDIX. AND/OR SIGN UP FOR A FREE ACCOUNT ONLINE AT KHANACADEMY.ORG. CHOOSE THE WORLD OF MATH MISSION, AND ACHIEVE PRACTICED LEVEL 1 ON THE FOLLOWING EXERCISES: 3. SE ASSESSMENT: REDO THE SE PRACTICE PROBLEMS #1- MAKING DIFFERENT CHOICES THAN YOU MADE THE FIRST TIME. (SOME A STUDENTS HAVE REPORTED REDOING SE PRACTICE PROBLEMS SEVERAL TIMES TO ACHIEVE SE READINESS.)
Trig Equations PS Sp016 Directions: Engage your Knowledge and Comprehension (KC) learning during exploration by actively listening for and committing here to a written record of the information that accurately responds to the following prompts. Memorize this information by reviewing it for homework every evening before the KC Assessment. Solving Trigonometric Equations 1. Give an example of a trigonometric equation, and identify the two trigonometric functions at whose intersections x-values balance this equation.. Give an example of a trigonometric equation that is not in standard form, and use properties of equality and simplification to re-write it in standard form. 3. Contextualize the coordinates at special right and degenerate positions on the unit circle in quadrant I. 4. Contextualize the coterminal angles in radians at special right and degenerate positions on the unit circle over two rotations counter-clockwise (positive angles) and one rotation clockwise (negative angles). 5. Give an example of a sequence of coterminal angles and name a trigonometric equation in standard form that has them as its solutions. 6. Give an example of a trigonometric identity, and state how this is different than a trigonometric equation. 7. Give an example of a trigonometric equation using the same nontrivial angles in each trigonometric expression, and re-write the equation renaming the angles as u. 8. State one of the Pythagorean identities, and use a property of equality and difference of squares to re-write it in four different ways. Re-written Using Property of Equality Re-written Using Difference of Squares 9. State one of the Sum identities and the Double-angle identity derived from it. 10. Give an example of a trigonometric equation using different angles in each trigonometric expression, and recognize an identity that could be used to re-write them as the same angle.
Lesson 3 Lesson Summary The sum and difference formulas for sine, cosine, and tangent are summarized below. For all real numbers and for which the expressions are defined, cos( ) = cos()cos() + sin()sin() cos( + ) =cos()cos() sin()sin() sin( ) = sin()cos() cos()sin() sin( + ) = sin()cos() + cos()sin() tan() tan() tan( ) = 1+tan()tan() tan() + tan() tan( + ) = 1 tan()tan(). Problem Set 1. Use the addition and subtraction formulas to evaluate the given trigonometric expressions. a. cos b. sin c. sin d. cos e. sin f. cos g. sin h. cos i. sin cos + cos sin j. sin cos cos sin k. sin cos + sin sin l. cos cos sin sin m. cos cos + sin sin n. sin cos cos sin Lesson 3: Addition and Subtraction Formulas S.18
Lesson 3. The figure below and to the right is obtained from the figure on the left by rotating the angle by about the origin. Use the method shown in Example 1 to show that cos( + ) = cos()cos() sin()sin(). 3. Use the sum formula for sine to show that sin( ) = sin()cos() cos()sin(). 4. Evaluate tan( + ) = () () tan() = () (). ()() to show tan( + ) =. Use the resulting formula to show that ()() 5. Show an( ) = ()() ()(). 6. Find the exact value of the following by using addition and subtraction formulas. a. tan b. tan c. tan d. tan e. f. g. Lesson 3: Addition and Subtraction Formulas S.19
Lesson 4 Lesson Summary The double-angle and half-angle formulas for sine, cosine, and tangent are summarized below. For all real numbers for which the expressions are defined, sin() =sin()cos() cos() = cos () sin () =cos () 1 =1sin () tan() = tan() 1 tan () sin =± 1 cos() cos =± cos() +1 tan cos () = ±1 1 + cos () Problem Set 1. Evaluate the following trigonometric expressions. a. sin cos b. sin cos c. 4sin cos d. cos sin e. cos 1 f. 1 sin g. cos h. i. j. cos k. cos Lesson 4: Addition and Subtraction Formulas S.3
Lesson 4 l. cos m. sin n. sin o. sin p. tan q. tan r. tan. Show that sin(3) =3sin()cos () sin (). (Hint: Use sin() =sin()cos() and the sine sum formula.) 3. Show that cos(3) = cos () 3sin ()cos(). (Hint: Use cos() = cos () sin () and the cosine sum formula.) 4. Use cos() = cos () sin () to establish the following formulas. a. cos () = () b. sin () = (). 5. Jamia says that because sine is an odd function, sin is always negative if is negative. That is, she says that for negative values of, sin = (). Is she correct? Explain how you know. 6. Ginger says that the only way to calculate sin is using the difference formula for sine since =. Fred says that there is another way to calculate sin. Who is correct, and why? 7. Henry says that by repeatedly applying the half-angle formula for sine we can create a formula for sin for any positive integer. Is he correct? Explain how you know. Lesson 4: Addition and Subtraction Formulas S.4
Lesson 6 Lesson Summary A wave is displacement that travels through a medium. Waves transfer energy, not matter. There are two types of waves: transverse and longitudinal. Sound waves are an example of longitudinal waves. When two or more waves meet, interference occurs and can be represented mathematically as the sum of the individual waves. The sum identity for sine is useful for analyzing the features of wave interference. Problem Set 1. Rewrite the sum of the following functions in the form ()+() = cos( + ). Graph = (), = (), and = () + () on the same set of axes. a. () =4sin(); () =3cos() b. () = 6sin(); () = 8cos() c. () = 3sin(); () =3cos() d. () = sin(); () = 7cos() e. () =3sin(); () = cos(). Find a sinusoidal function () = sin( + ) + that fits each of the following graphs. a. b. Lesson 6: Waves, Sinusoids, and Identities S.36
Lesson 6 c. d. 3. Two functions and are graphed below. Sketch the graph of the sum +. a. Lesson 6: Waves, Sinusoids, and Identities S.37
Lesson 6 b. c. d. e. Lesson 6: Waves, Sinusoids, and Identities S.38
Lesson 1 Problem Set 1. Solve the following equations. Approximate values of the inverse trigonometric functions to the thousandths place, where refers to an angle measured in radians. a. 5=6cos() b. 1 =cos +1 4 c. 1=cos3( 1) d. 1. = 0.5 cos() + 0.9 e. 7=9 cos() 4 f. =3sin() g. 1 =sin 1 1 4 h. =3sin(5 +) + i. = () j. cos() = sin() k. sin (cos()) = 3 l. tan() =3 m. 1 =tan(5 +) 3 n. 5=1.5 tan() 3. Fill out the following tables. sin () cos () sin () cos () 1 0 3 1 1 3 1 Lesson 1: Inverse Trigonometric Functions S.88
Lesson 1 3. Let the velocity in miles per second of a particle in a particle accelerator after seconds be modeled by the function = tan 6000 on an unknown domain. a. What is the -value of the first vertical asymptote to the right of the -axis? b. If the particle accelerates to 99% of the speed of light before stopping, then what is the domain? Note: 186000. Round your solution to the ten-thousandths place. c. How close does the domain get to the vertical asymptote of the function? d. How long does it take for the particle to reach the velocity of Earth around the sun (about 18.5 miles per second)? e. What does it imply that is negative up until = 3000? Lesson 1: Inverse Trigonometric Functions S.89
Pre-Calculus Addendum 441 to Lesson 1 Solving Solving Trigonometric Equations Worksheet Part II Solve each equation over the interval [0, ). 1. cos x 4 5. sin x 1 0 3. tan x 3 0 4. 5cos x 3 3cos x 5. 4csc x 0 6. sin x5cos x 5
Solve each equation. Give a general solution. 7. 4sin x 0 8. 3tan x 3 0 9. 3x sec 0 10. sin x4sin x5 0 11. cot xsec xcot x 0 1. 5cosx1 3cosx Solve each equation over the interval (, ). 13. 16cos x 8 0 14. cos x3cos x1 0
Trig Equations SE Practice Sp016 1. Construct a trigonometric system of two equations (say, g and h) in two variable that has the following properties, and evaluate how analytically: a. has a solutions at x = π 4 b. the transformed system g( x) and h( x) has 4 solutions on [ 0,π ) c. squaring both sides of the equation g( x) = h( x) yields 4 solutions on [ 0,π ) d. the sum of the extraneous solutions generated in part c. is 5π.. Construct a trigonometric system of two equations (say, g and h) in two variable that has the following properties, and evaluate how analytically: a. has a solution at x = 5π 4 b. the transformed system g( x) and h( x) has 3 solutions on [ 0,π ) c. squaring both sides of the equation g( x) = h( x) yields 4 solutions on 0,π [ ) d. the distance between the extraneous solutions generated in part c. is π.