UNIT ONE ADVANCED TRIGONOMETRY MATH 611B 15 HOURS Revised Feb 6, 03 18
SCO: By the end of grade 1, students will be expected to: B10 analyse and apply the graphs of the sine and cosine functions C39 analyse tables and graphs of sine and cosine equations to find patterns Elaborations - Instructional Strategies/Suggestions Relating Graphs and Solutions (5.1) Invite student groups to do the Getting Started and Warm Up exercises on p.4-3 as selected in the Suggested Resources column. Challenge student groups to do the Explore and Inquire on p.44. Student groups should read and discuss p.45-47 Note To Teachers: This section is essentially systems of equations composed of a sinusoidal and linear equation. The students might develop a better understanding of the patterns occurring if they do the Relating Graphs and Solutions Worksheet (5.1) at the end of the unit. It may be easier for students to have the graphing calculator in the degree mode to do the worksheet. 19
Worthwhile Tasks for Instruction and Assessment Suggested Resources Relating Graphs and Solutions (5.1) Technology Graph using a graphing calculator. Inductively find a formula for the exact solutions: 1 a) sin x = b) sin x = 1 Solution for (b): Relating Graphs and Solutions (5.1) Relating Graphs and Solutions Worksheet (5.1) Math Power 1 p.47 # 5,7,9 Applications p.48 # 5, 8 0
SCO: By the end of grade 1, students will be expected to: B11 derive, analyse and apply angle and arclength relationships using the unit circle C4 create and solve trigonometric equations C48 solve trigonometric equations with and without graphing technology Elaborations - Instructional Strategies/Suggestions Solving Trig Equations (5.) Student groups should do the Explore and Inquire on p.49. Student groups should review the concepts of: < angles in standard position < reference angles < ratios for special angles 0, 30, 45, 60 and 90 Invite student groups to read and discuss the examples on p.49-5. Students will be expected to solve these problems both algebraically and graphically. Students should be exposed to two graphical methods for the following example: Find the solutions for cosx!1 = 0 for!b # x # B. Method 1: nd Calc 5:intersect Method : nd Calc: :zero So students should see that they can find the solutions by looking for the intersection of two graphs or by looking for the zeros of the single graph. 1
Worthwhile Tasks for Instruction and Assessment Suggested Resources Solving Trig Equations (5.) Pencil/Paper Solve sin + 1 =.5 algebraically and graphically for, 0 # # B. Group Activity Solve for x where 0 # x # B. Then give a general solution for: 3sin x + sin x = 1. Solving Trig Equations (5.) Math Power 1 p.5 #1-31 odd, 10, 18,0 Applications p.53#3(a),33(b),35(b) Trig Equation Worksheet (5.)
SCO: By the end of grade 1, students will be expected to: B10 analyse and apply the graphs of the sine and cosine functions C48 solve trigonometric equations with and without graphing technology Elaborations - Instructional Strategies/Suggestions Using Technology (5.3) Student groups should do the Explore and Inquire on p.54. Student groups should read and discuss the examples on p.55-56. The text does not make clear the fact that there are two graphical methods that can be used. Looking at the example on p.54 we can use: a) systems of equations: graph y = cos x, y = x and find the intersection point. b) combine both functions into a single function y = cos x! x and find the zeros of the function. Note to Teachers: This function could have been written as y = x! cos x yielding the same zero or solution. 3
Worthwhile Tasks for Instruction and Assessment Suggested Resources Using Technology (5.3) Technology A ferris wheel has a diameter of 50m and turns at a rate of 1.5 revolutions per minute. The height of a seat above the ground after t minutes can be described using h = 1! 5 cos 3Bt. How long after the ride starts will your seat be 31 m off the ground for the first time. Using Technology (5.3) Math Power 1 p.56 #1-5, 8 Technology Solve for, 0# # B, the equation sin θ + 1 = 05. Technology The shown below shows y = cos and y =.5x! 1 using the window [!B,B,B/4] and [!,,1]. a) To what single equation does the graph provide the solution? b) Use the graph to give an approximate solution. 4
SCO: By the end of grade 1, students will be expected to: B1 derive and apply the Reciprocal and Pythagorean Identities Elaborations - Instructional Strategies/Suggestions Trigonometric Identities (5.4) Student groups should do the Explore & Inquire on p.58. They should then read and discuss p.59-63. Students should appreciate the difference between an equation and an identity. An equation is a statement that is true for a limited number of values. An identity is a statement that is true for any value of the variable. This section uses the basic trig identities below to solve various problems: Pythagorean Identities Quotient Identities tan θ = sin θ cos θ sin tan θ + cos θ = 1 θ + 1 = 1 + cot θ = sec csc Reciprocal Identities sin θ csc θ = 1 cos θ sec θ = 1 tan θ cot θ = 1 θ θ cot θ = cos θ sin θ Suggestions for verifying Trig Identities: < work with the more complicated side of the equation < substitute one or more of the basic identities to simplify, factor or multiply to simplify < multiply expressions equivalent to one < express trig functions in terms of sine and cosine The properties of equalities do not apply, so that operations cannot be performed on both sides of an unverified identity. 5
Worthwhile Tasks for Instruction and Assessment Suggested Resources Trigonometric Identities (5.4) Group Activity Determine whether or not the following is a trigonometric 7 sin θ + 5cos θ identity: = 7 sec θ + 5csc sin θ cos θ θ Trigonometric Identities (5.4) Math Power 1 p.64 #1-31 odd omit # 7 Trig Identity Worksheet (5.4). Performance Demonstrate to your group/class the following. When an object is fired with an initial velocity v 0 at an angle of elevation, its height y above the ground and its horizontal displacement x are related by the equation: gx x sin θ y = + v0 cos θ cos θ Rewrite this equation so that tan is the only trig function appearing. 6
SCO: By the end of grade 1, students will be expected to: B13 explore and verify other trigonometric identities and solve trigonometric equations B41 derive and apply the compound angle identities and the half and double angle identities Elaborations - Instructional Strategies/Suggestions Sum, Difference, Double Angle Identities (5.5) Challenge students to do the Investigation on p.66-7. Students will not be expected to memorize the following identities. Sum and Difference Identities sin(a + B) = sina cosb + cosa sinb sin(a! B) = sina cosb! cosa sinb cos(a + B) = cosa cosb! sina sinb cos(a! B) = cosa cosb + sina sinb tan(a + B) = tana + tanb 1! tana tanb tan(a! B) = tana! tanb 1 + tana tanb Double Angle Identities sina = sina cosa cosa = cos A! sin A or cos A! 1 or 1! sin A tana = tana where A ± B/4 & B/ + nb 1! tan A 7
Worthwhile Tasks for Instruction and Assessment Suggested Resources Sum, Difference, Double Angle Identities (5.5) Pencil/Paper Use the sum or difference identity for tangent to find the exact value of tan 85. Activity Use the sum or difference identity for cosine to find the exact value of cos 735. Performance Fπ 3 Verify that csc sec is an identity. HG I + A K J = A Group Activity If sin = /3, and has its terminal side in the first quadrant, find the exact value of each function: a) sin Sum, Difference, Double Angle Identities (5.5) Math Power 1 p.7 #1-7 odd, Double Angle Worksheet (5.5) Applications p.73 #37,39,40,41, 43(d),47 Enrichment Applications Worksheet 5.5 at the end of the unit. b) cos c) tan d) cos 4 8
Relating Graphs and Solutions Worksheet (5.1) Graph each of the following using a graphing calculator. Inductively, find a formula for the exact solution: a) y = sin x ; y = 0 b y = sinx ; y = 0 c) y = sin3x ; y = 0 d) y = sin x ; y = 1 e) y = sin x ; y = 1 f) y = sin 3x ; y = 1 g) y = cos x ; y = 0 h) y = cos x ; y = 0 i) y = cos 3x ; y = 0 j) y = cos x ; y = 1 k) y = cos x ; y = 1 l) y = cos 3x ; y = 1 9
Solutions for Worksheet (5.1) Window settings a) y = sin x ; y = 0 nb; 0, 180 [0,B,B],[!3,3,1] b y = sinx ; y = 0 nb/ 0, 90, 180 [0,4B,B/],[!3,3,1] c) y = sin3x ; y = 0 nb/3 60, 10, 180 [0,4B,B/3],[!3,3,1] d) y = sin x ; y = 1 B/ + nb 90, 450 [0,4B,B/],[!3,3,1] e) y = sin x ; y = 1 B/4 + nb 45, 5 [0,B,B/4],[!3,3,1] f) y = sin 3x ; y = 1 B/6 + nb/3 30, 150, 70 [0,B,B/6],[!3,3,1] g) y = cos x ; y = 0 B/ + nb 90, 70 [0,4B,B/],[!3,3,1] h) y = cos x ; y = 0 B/4 + nb/ 45, 135, 5 [0,B,B/4],[!3,3,1] i) y = cos 3x ; y = 0 B/6 + nb/3 30, 90, 150 [0,B,B/6],[!3,3,1] j) y = cos x ; y = 1 nb 0, 360 [0,4B,B/],[!3,3,1] k) y = cos x ; y = 1 nb 0, 180, 360 [0,B,B/],[!3,3,1] l) y = cos 3x ; y = 1 nb/3 0, 10, 40 [0,B,B/3],[!3,3,1] 30
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Trig Equation Worksheet (5.) Solve for x: 1. sin x cos x = 0,. sin x + sin x = 0, 3. cos x - 3 sin x = 1, 4. tan x cos x - cos x = 0, 5. sin x sin x = cos x, state the general solution 6. cos x + 3 sin x - 3 = 0, 7. cos x + cos x - 3 = 0, 8. cos x + sin x = 0, 9. tan x - 3 sec x + 3 = 0, 10. cos x + tan x = 0, 3
Projectile Motion Activity (5.3) OBJECT: Explore mathematically the time in the air, the maximum height achieved, and the horizontal range of an object launched at various angles. PROCEDURE: An object is launched at an initial speed of 30 m/s at the following angles. Use Parametric Graphing to complete the following table. t max height t total y (height) x (range) 10 0 0 0 30 0 40 0 45 0 50 0 60 0 70 0 80 0 90 0 Questions: At what angle will the object remain in the air for the longest time? At what angle will the object have the largest horizontal range? 33
or animate Press graph then trace These last two screens show the time to reach the maximum height, the maximum height, the time to fall back to Earth and the total distance travelled. 34
Prove each of the following to be identities. Trig Identity Worksheet (5.4) 1. 1 tan θ 1 + = cos θ sec θ 1. sin A+ tan A 1+ seca = sin A 3. 4. 5. 1 sin θ = sin θ csc θ 1 1+ tana sin A+ cos A= sec A tan x sin x = sin x tan x Determine if the following are identities. 6. cot x+ cos x = cos x(1 + sin x) sin x 7. cotθ + cosθ = cosθ sinθ 8. cot x + cos x = tan x+ sin x 35
Double Angle Worksheet (5.5) Prove each of the following to be identities: 11. 1. 13. 14. 15. 16. 17. 18. 19. 0. 1. Simplify each of the following: 1.. 3. 4. 5. 36
Applications Worksheet (5.5) 1. Have you ever tried to tune in a radio station only to have it fade in and out or to have interference from other channels disrupt your listening pleasure. This is called destructive or constructive interference. What type of interference results when the following two signals are combined? y = 0 sin(3t + 45 ) and y = 0 sin(3t + 10 ) TI-83 in deg mode; window dimensions [0,360,30],[!40,40,8] graph Y 1,Y and Y 3 = Y 1 + Y In an electric circuit containing a capacitor, inductor and a resistor the voltage drop across the inductor is given by V L = I 0 TL cos(tt + B/), where I 0 is the peak current, T is the frequency, L is the inductance, and t is the time. Use the sum identity for cosine to express V L as a function of sin Tt. 3. Water fountains many times have water jets that shoot water into the air to create parabolic arcs. When a stream of water is shot into the air at an angle of with the horizontal, then water will travel a v horizontal distance of D = sin θ and reach a maximum height of g where g is the acceleration due to gravity. H a) as a function in simplest terms. D H = v g sin θ b) What is the ratio of the maximum height of the water to the horizontal distance it travels for an angle of 7? 4. An AC circuit consists of a power supply and resistor. If the current in the circuit at time t is I 0 sintt, then the power delivered to the resistor is P = I 0 R sin Tt, where R is the resistance. Express the power in terms of cos Tt. 5. The index of refraction for a medium through which light passes is the ratio of the velocity of light in a vacuum to the velocity of light in the medium. For light passing through a prism the index of refraction is 1 sin ( α + β ) n = where " is the deviation angle and $ is the angle of the apex of the prism. β sin If $ = 60, show that n = 3 sin + cos. Answers α α 37
1. destructive interference. VL = I0 ω L sin ω t 3. H a) D = 1 4 tan θ b) H D = 1 o tan 7 4 013. 1 1 4. P = I0 R I0 R cos ωt 5. n = = = 1 sin[ ( α + β) β sin 1 o sin[ ( α + 60 ) o sin 30 1 o 1 sin α cos 30 + cos α sin 30 1 3 α 1 α = ( sin + cos ) α 1 α = 3 sin + cos o 38