1 General Outcome: Develop trigonometric reasoning. Specific Outcomes: Unit 3 Trigonometry 3.1 Demonstrate an understanding of angles in standard position, expressed in degrees and radians. 3. Develop and apply the equation of the unit circle. 3.3 Solve problems, using the six trigonometric ratios for angles expressed in radians and degrees. 3.4 Graph and analyze the trigonometric functions sine, cosine, and tangent to solve problems. y = asin b( x x) + d y = a cos b( x x) + d 3.5 Solve, algebraically and graphically, first and second degree trigonometric equations with the domain expressed in degrees and radians. 3.6 Prove trigonometric identities using: Topics: reciprocal identities quotient identities Pythagorean identities sum or difference identities (restricted to sine, cosine, and tangent) double-angle identities (restricted to sine, cosine, and tangent) Trigonometry Fundamentals (Outcomes 3.1 & 3.3) Page Unit Circle (Outcomes 3. & 3.3) Page 4 Graphing Sine & Cosine Functions (Outcome 3.4) Page 3 Applications of Sine & Cosine Functions (Outcome 3.4) Page 51 Graphs of Other Trigonometric Functions (Outcome 3.4) Page 6 Solving Trigonometric Equations Graphically (Outcome 3.5) Page 66 Solving Trigonometric Algebraically (Outcome 3.5) Page 70 Trigonometric Identities (Outcome 3.6) Page 81 Sum & Difference Identities (Outcome 3.6) Page 91 Solving Trigonometric Equations Part II (Outcome 3.5) Page 101
Unit 3 Trigonometry Trigonometry Fundamentals: Standard Position: An angle is in standard position when its vertex is at the origin and its initial arm is on the positive x-axis. Angles in Standard Position: Angles not in Standard Position: Angles measured counterclockwise are positive and angles measured clockwise are negative.
3 Ex) Sketch the following angles. a) 130 b) 30 c) 490 **Angles that have the same terminal arm are called Coterminal Angles. Ex) List 5 angles that are coterminal with 60 **The smallest positive representation of an angle is called the Principal Angle.
4 Ex) Determine the principal angle for each of the following. a) 1040 b) 713 c) 51 Radian Measure: Radian measure is a different way in which to measure an angle. We could measure the distance to Edmonton in km or miles, meaning we could measure the distance using two different systems of measurement where 1 km 1 mile. When measuring angles there are different systems, we can measure angles in degrees or we can measure them in radians. Note: 1 1 rad. Radian measure is a ratio of a circles arc length over its radius a = r
5 Ex) What is the measure of an angle that is 360 in radians? rad = 180 Ex) Convert the following to radians. a) 90 b) 45 c) 150 d) 10 e) 1 f) 1080
6 Ex) Convert the following to degrees. a) 3 b) 5 8 c) 15 d) 6 7 e) 5.4 f) 11.7 Ex) Determine the measure of x.
Ex) Determine one positive and one negative coterminal angle for the following. 7 a) 3 4 b) 7 1 Ex) Determine the principal angle in each case. a) 9 6 b) 11 3
8 Special Angles: Determine the coordinates (height and how far over) of the sun for each of the following cases. Thus: sin 45 = & cos45 = Thus: sin60 = & cos60 =
9 Thus: sin30 = & cos30 = **Summary** sin30 = cos30 = sin 45 = cos45 = sin60 = cos60 =
10 Quadrants: CAST Rule: Reciprocal Trigonometric Functions: sin = opp hyp cos = adj hyp tan = opp adj tan = sin cos **New** csc = sec = cot =
11 Reference Angles: A reference angle is the angle between the terminal arm and the horizon or x-axis. Reference angles are always between 0 and 90. Ex) Determine the six exact primary trigonometric ratios for each of the following. a) 40
1 b) 4 Ex) The point ( ) 5, 7 lies on the terminal arm of angle. Determine the six exact primary trigonometric ratios for.
3 Ex) If sin = and cos is negative, determine the other 7 5 exact primary trigonometric ratios for. 13 8 Ex) If sec = and tan 0, determine the exact value of 5 sin.
14 Trigonometry Fundamentals Assignment: 1) Convert each of the following to radians. Express your answer as an exact value and as an approximate to the nearest hundredth. a) 30 b) 300 c) 1 d) 90 e) 750 f) 135 ) Convert each of the following to degrees. Round your answer to the nearest hundredth if necessary. a) 3 b) 5 4 c) 3 d).75 e) 1 5 f) 1
15 3) In which quadrant do each of the following angles terminate? a) 650 b) 1 c) 19 d) 11 3 e) 5 f) 8.5 4) Determine one positive and one negative coterminal angle for each of the following angles. a) 7 b) 11 7 c) 05 d) 9. e) 50 f) 14 3
16 5) Determine the value of the indicated variable for each of the following cases below. Round your answers to the nearest hundredth of a unit. a) b) c) d)
17 6) A rotating water sprinkler makes one revolution every 15 seconds. The water reaches a distance of 5 m from the sprinkler. a) What angle in degrees does the sprinkler rotate through in 9 seconds? b) What is the area of sector watered in 9 seconds? 7) Complete the table shown below by converting each angle measure to its equivalent in the other systems. Round your answers to the nearest tenth where necessary. Revolutions Degrees Radians 1 rev 0.7 rev 3.5 rev 70 40 460 5 6 1.7 3 8
18 8) Determine the six primary trigonometric ratios for each of the following. Leave answers as exact values. a) 45 b) 40 c) 5 6 d) 3
19 9) In with quadrant(s) may terminate under the following conditions? a) cos 0 b) tan 0 c) sin 0 d) sin 0 e) cos 0 f) sec 0 & cot 0 & csc 0 & tan 0 10) Express the given quantity using the same trigonometric ratio and its reference angle. For example, cos110 = cos70. For angle measures in radians, give cos3 = cos 3. exact answers. For example, ( ) a) sin 50 b) tan 90 c) sec135 d) cos4 e) csc3 f) cot 4.95
0 11) Determine the exact value of each expression. a) cos60 + sin 30 b) ( sec45 ) c) 5 5 cos sec 3 3 d) ( tan 60 ) + ( sec60 ) e) 7 7 cos + sin 4 4 f) 5 cot 6 1) Determine the exact measure of all angles that satisfy the following. a) 1 sin =, where 0
1 b) cot = 1, where c) sec =, where 180 90 d) ( cos ) 1 =, where 360 360 13) Determine the exact values of the other five trigonometric ratios under the given conditions. a) 3 sin =, where 5
b) cos =, where 3 3 c) tan =, where 70 360 3 d) 4 3 sec =, where 70 180 3
14) The point ( 3, 4) lies on the terminal arm of angle. Determine the six exact primary trigonometric ratios for. 3 15) The point ( 5, ) lies on the terminal arm of angle. Determine the six exact primary trigonometric ratios for. 16) The point ( ) 5, 1 lies on the terminal arm of angle. Determine the six exact primary trigonometric ratios for.
4 Unit Circle: The unit circle is a circle with a radius of 1 unit whose center is located at the origin. Equation of the unit circle: x + y = 1 Consider a point on the unit circle( x, y ) to be a point on the terminal arm of cos = sin = tan =
5
6 Ex) Determine the exact value for the following. a) sin 40 b) 7 cos 4 c) 5 sec 6 d) cot540 e) 4 csc 3 f) tan 150 Ex) Solve the following for x. a) sin x =, where 0 x 360
7 b) cos 0.8090 x =, where 0 x 360 c) csc x =, where 0 x 3 d) tan x is undefined e) 4cos x = 3, where 0 x
8 Unit Circle Assignment: 1) Determine whether or not each of the following points is on the units circle. a) 3 1, 4 4 b) 5 1, 13 13 c) 5 7, 8 8 d) 4 3, 5 5 e) 3 1, f) 7 3, 4 4 ) Determine the coordinate for all points on the unit circle that satisfy the following conditions. a) 1, 4 y in quadrant I b) x, in quadrant II 3
9 c) 7, y 8 in quadrant III d) 5 x, 7 in quadrant IV e) 1 x, 3 where x 0 f) 1, 13 y not in quadrant I 3) If P( ) is the point at the intersection of the terminal arm of angle and the unit circle, determine the exact coordinates of each of the following. a) P( ) b) P c) P 3 d) 7 P 4 e) 5 P f) 5 P 6
30 4) If is in standard position and 0, determine the measure of if the terminal arm of goes through the following points. a) ( 0, 1) b ) 1 1, c) 1 3, d) 3 1, e) ( 1, 0 ) f) 3 1, 5) Determine one positive and one negative measure of if 3 1 P( ) =,.
31 3 6) The point P( ) =, y lies on the terminal arm of an angle in standard 5 position and on the unit circle. P( ) is in quadrant IV. a) Determine the value of y. b) What is the value of tan? c) What is the value of csc?
3 Graphing Sine and Cosine Functions: Graph of y = sin Complete the following table y y 0 30 60 90 10 150 180 = sin x 10 40 70 300 330 360 = sin x Graph y = sin
33 Graph of y = cos Complete the following table y 0 = cos x 6 3 5 3 6 y 7 4 3 5 11 6 3 3 6 = cos x Graph y = cos
34 Transformations to y = sin x & y = cos x : y = asin b( x c) + d Amplitude: Changing the parameter a in y = asin b( x c) + d will affect the amplitude of the graph. A = amplitude *if a is negative, the amplitude is a and the graph is reflected about the x-axis
35 Period: Changing the parameter b in y = asin b( x c) + d will affect the period of the graph Period 360 = or b Period = b *If b is negative, b is used to find the period of the graph, the fact that b is negative reflects the graph about the y-axis. **Like in Unit 1 the equation must be in the form y = asin b( x c) + d not y = asin( bx c) + d in order to properly see the period and phase shift.
36 Phase Shift: Changing the parameter c in y = sin b( x c) + d affects the phase shift of the graph (moves the graph left or right). *Think c determines where we begin drawing the sine or cosine pattern sine begins cosine begins Vertical Displacement: Changing the parameter d in y = sin b( x c) + d affects the vertical displacement of the graph (moves the graph up or down). *Think d location of the median line
37 Ex) Graph y ( x ) = 3cos 10 Ex) Graph y 1 ( ) = 4sin + 3 6 Ex) Graph y ( x ) = 4cos3 3
38 Ex) Graph y ( x ) = sin 45 + 1.5 Ex) Graph y = 4cosx 4 3
39 Ex) Determine the equation of a sine function with an amplitude of 3 and a period of. 6 Ex) Determine the equation of a cosine function with an amplitude of 3 and a period of 70. 3 7 Ex) If f( x) = 14sin + 50 4 5 determine the following. a) The maximum of b) The period of the graph the graph c) The location of the first minimum found to the right of the y-axis
Ex) Determine both the sine and the cosine equation for each of the following. a) 40 b)
41 c) d)
4 e) f)
43 Graphing Sine and Cosine Functions Assignment: 1) Determine the amplitude and the period, in both degrees and radians, for the graphs of each of the following. a) y sin( 6 ) = b) 1 1 gx ( ) = cos 3 3 c) 3 y = 16cos d) 6 f( x) = 15sin 5 ) Determine the range of each function. a) y= 3cosx + 5 b) y ( x ) = sin + 3 = + c) y cos( x 50 ) d) y 1.5sin x 4 3 = + + 3 4
44 3) Match each function with its graph. i) y = 3cos ii) y = cos3 iii) y = sin iv) y = cos a) b) c) d) 4) Determine the equation of the cosine curve that has a range given by y 6 y 4, y Rand consecutive local maximums at, 4 9 and 8, 4 9.
45 5) Match each function with its graph. i) y = sin 4 ii) y = sin + 4 iii) y = sin 1 iv) y = sin + 1 a) b) c) d) 6) Determine the equation of the sine curve that has local minimum at ( 15, 13) and a local maximum that immediately follows it at ( 35, 19 ). (Note: There are no other local maximum or minimums between the two given points.)
46 7) Graph each of the following. 3 = sin ( 60 ). 4 a) y ( ) 4 = 4cos 135 + 3 b) y ( )
47 c) 5 7 y = cos 3 4 d) 7 1 3 y = sin 4 + 3 4
48 8) If y = f ( x) has a period of 6, determine the period of 1 y = f x. 9) If sin 0.3 =, determine the value of sin sin( ) sin( 4 ) + + + +. 10) Determine both a sine and cosine equation that describes of the graphs given below. a)
49 b) c)
50 d) 11) Determine the value of a to make each statement below true. a) 4sin( x 30 ) = 4cos( x a) 4 b) sin x = cos( x a) c) 3cos x = 3sin( x + a) ( ) d) cos( x + 6 ) = sin ( x + a)
51 Applications of Sine and Cosine Functions: Ex) By finding the averages of high and low tide, the depth of water dt () in meters, at a sea port can be approximated using the sine function d( t) =.5sin 0.164 ( t 1.5) + 13.4 where t is the time in hours. a) Sketch the graph of this function. b) What is the period of the tide (length of time from one low tide to the next low tide). c) A cruise ship needs a depth of at least 1 m of water to dock safely. For how many hours per tide cycle can the ship safely dock?
5 Ex) The table below shows the average monthly temperatures for Winnipeg. Month (m) Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec 1 3 4 5 6 7 8 9 10 11 1 Temperature (t) -19-16 -8 3 11 17 0 18 1 6-5 -14 a) Graph this data. b) Crate a cosine equation that describes this data. c) Using the equation created, predict what the temperature will be on April 15 th.
Ex) A nail is caught in the tread of a rotating tire at point N in the following picture. 53 The tire has a diameter of 50 cm and rotates at 10 revolutions per minute. After 4.5 seconds the nail touches the ground for the first time. a) Indicate the proper scale on the horizontal and vertical axis for the graph above. b) Determine the equation for the height of the nail as a function of time. c) How far is the nail above the ground after 6.5 seconds? (Round your answer to the nearest tenth.)
Ex) A ferris wheel has a diameter of 76 m and has a maximum height of 80 m. If the wheel rotates every 3 minutes, draw a graph that represents the height of a cart as a function of time. Assume the cart is at its highest position at t = 0. Show three complete cycles. 54 Determine the cosine equation that describes this graph. How many seconds after the wheel starts rotating does the cart first reach a height of 10 m? (Round your answer to the nearest second.)
55 Applications of Sine & Cosine Functions Assignment: 1) The alarm in a noisy factory is a siren whose volume, V decibels fluctuates so that t seconds after starting, the volume is given by the function V ( t) = 18sin t + 60. 15 a) What are the maximum and minimum volumes of the siren? b) Determine the period of the function. c) Write a suitable window which can be used to display the graph of the function. d) After how many seconds, to the nearest tenth, does the volume first reach 70 decibels? e) The background noise level in the factory is 45 decibels. Between which times, to the nearest tenth of a second, in the first cycle is the alarm siren at a lower level than the background noise? f) For what percentage, to the nearest per cent, of each cycle is the alarm siren audible over the background factory noise?
56 ) A top secret satellite is launched into orbit from a remote island not on the equator. When the satellite reaches orbit, it follows a sinusoidal pattern that takes is north and south of the equator, (ie. The equator is used as the horizontal axis or median line). Twelve minutes after it is launched it reaches the farthest point north of the equator. The distance north or south of the equator can be represented by the function d( t) = 5000cos ( t 1) 35 where d (t) is the distance of the satellite north or the equator t minutes after being launched. a) How far north or south of the equator is the launch site? Answer to the nearest km. b) Is the satellite north or south of the equator after 0 minutes? What is the distance to the nearest km? c) When, to the nearest tenth of a minute, will the satellite first be 500 km south of the equator?
57 3) The height of a tidal wave approaching the face of the cliff on an island is represented by the equation h( t) = 7.5cos t 9.5 where h (t) is the height, in meters, of the wave above normal sea level t minutes after the wave strikes the cliff. a) What are the maximum and minimum heights of the wave relative to normal sea level? b) What is the period of the function? c) How high, to the nearest tenth of a meter, will the wave be, relative to normal sea level, one minute after striking the cliff? d) Normal sea level is 6 meters at the base of the cliff. i) For what values of h would the sea bed be exposed? ii) How long, to the nearest tenth of a minute, after the wave strikes the cliff does it take for the sea bed to be exposed? iii) For how long, to the nearest tenth minute, is the sea bed exposed?
58 4) The depth of water in a harbour can be represented by the function d( t) = 5cos t + 16.4 6 where d (t) is the depth in meters and t is the time in hours after low tide. a) What is the period of the tide? b) A large cruise ship needs at least 14 meters of water to dock safely. For how many hours per cycle, to the nearest tenth of an hour, can a cruise ship dock safely? 5) A city water authority determined that, under normal conditions, the approximate amount of water, W (t), in millions of liters, stored in a reservoir t months after May 1, 003, is given by the formula W( t) = 1.5 sin t. 6 a) Sketch the graph of this function over the next three years. b) The authority decided to carry out the following simulation to determine if they had enough water to cope with a serious fire. If on November 1, 004, there is a serious fire which uses 300 000 liters of water to bring under control, will the reservoir run dry if water rationing is not imposed? If so, in what month will this occur?
59 6) The graph shows the height, h meters, above the ground over time, t, in seconds that it takes a person in a chair on a Ferris wheel to complete two revolutions. The minimum height of the Ferris wheel is meters and the maximum height is 0 meters. a) How far above the ground is the person as the wheel starts rotating? b) If it takes 16 seconds for the person to return to the same height, determine the equation of the graph in the form h( t) = asinbt + d c) Find the distance the person is from the ground, to the nearest tenth of a meter, after 30 seconds. d) How long from the start of the ride does it take for the person to be at a height of 5 meters? Answer to the nearest tenth of a second.
60 7) A Ferris wheel ride can be represented by a sinusoidal function. A Ferris wheel at Westworld Theme Park has a radius of 15 m and travels at a rate of six revolutions per minute in a clockwise rotation. Ling and Lucy board the ride at the bottom chair from a platform one meter above the ground. a) Sketch three cycles of a sinusoidal graph to represent the height of Ling and Lucy are above the ground, in meters, as a function of time, in seconds. b) Determine the equation of the graph in the form ( ) cos ( ) h t = a b t c + d. c) If the Ferris wheel does not stop, determine the height Ling and Lucy are above the ground after 8 seconds. Give your answer to the nearest tenth of meter. d) How long after the wheel starts rotating do Ling and Lucy first reach 1 meters from the ground? Give your answer to the nearest tenth of a second. e) How long does it take from the first time Ling and Lucy reach 1 meters until they next reach 1 meters from the ground? Give your answer to the nearest second.
61 8) Andrea, a local gymnast, is doing timed bounces on a trampoline. The trampoline mat is 1 m above ground level. When she bounces up, her feet reach a height of 3 m above the mat, and when she bounces down her feet depress the mat by 0.5 m. Once Andrea is in a rhythm, her coach uses a stopwatch to make the following readings: At the highest point the reading is 0.5 seconds. At the lowest point the reading is 1.5 seconds. a) Sketch two periods of the graph of the sinusoidal function which represents Andrea s height above the ground, in meters, as a function of time, in seconds. b) How high was Andrea above the mat when the coach started timing? c) Determine the equation of the graph in the form h( t) = asinbt + d. d) How high, to the nearest tenth of a meter, was Andrea above the ground after.7 seconds? e) How long after the timing started did Andrea first touch the mat? Answer to the nearest tenth of a second.
6 Graphs of Other Trigonometric Functions: Use information about the graph of y tan y = sin and y cos = and y = cot = to construct y = tan y = cot Period: Period: Domain: Domain: Range: Range:
63 1 Use the knowledge that csc x = sin x and sec x create the graph of y= csc x and y= sec x. = 1 cos x to Period: Period: Domain: Domain: Range: Range:
64 Ex) Determine the period, domain, and range of y = 3sec x+. Ex) Determine the period, domain, and range of y = csc3x 1. Ex) Determine the period, domain, and range of y= tan( x 3 )
65 Graphs of Other Trigonometric Functions Assignment: 1) Determine the Period, Domain, and Range for the graph given by y = 3sec + 7. ( ) ) Determine the Period, Domain, and Range for the graph given by 1 y = tan 4. 3 3) Determine the Period, Domain, and Range for the graph given by y = 1csc ( 40 ) + 5. 3 ( ) 4) Identify the restrictions on f( x) = cot 3( 60 )
66 Solving Trigonometric Equations Graphically: We can solve equations by graphing each side of the equation and finding the intersection Ex) Solve sin = 1 Graph Window y1 sin( x) 0.5 = x: [-360, 70, 90] y = y: [-1.5, 1.5, 1] Interval Notation: Interval notation gives us another way to represent solution sets. Ex)
67 Ex) Solve 4cos = 3 for 0, then state the general solution. Ex) Solve sin x+ sin x 1= 0, for 0 x 360 and state the general solution. Ex) Solve 3sec + 11 = 5, for 0, then state the general solution.
68 Solving Trigonometric Equations Graphically Assignment: 1) Without solving, determine the number of solutions for each trigonometric equation in the specified domain. Explain your reasoning. a) 3 sin =, 0 b) 1 cos =, b) tan = 1, 360 180 d) 3 sec =, 0 3 1 5 ) The equation cos =, 0, has solutions and. Suppose the 3 3 domain is not restricted. a) What is the general solution corresponding to =? 3 5 b) What is the general solution corresponding to =? 3
69 3) Solve each equation for 0. Give solutions to the nearest hundredth of a radian. a) tan = 4.36 b) sin = 0.91 c) sec =.77 d) csc = 1.57 4) Solve each equation in the specified domain. a) 3cos 1= 4cos, 0 b) sin x 1= 0, 360 x 360 c) 3 tan + 1= 0, d) 3sec x + = 0, 3 5) Explain why the equation sin 0 = has no solution in the interval (, ).
70 Solving Trigonometric Equations Algebraically: When solving trigonometric equations we must remember that there are usually principal solutions ( solutions within 360 or ). Tools for Solving: S T A C sin30 sin 45 sin60 = = = 1 3 cos30 cos45 cos60 = = = 3 1 tan = sin cos cot = cos sin csc = 1 sin sec = 1 cos *understand how reference angles can be used y = sin x y= cos x
71 Level I: Isolate the trigonometric function, then use your knowledge of reference angles to find solutions Ex) Solve each of the following. a) 3 sin =, 0 360 and the general solution b) cos 1= 0, 0 and the general solution c) sin 0 + =, ) 0, and the general solution
7 d) 1 cos 3cos + =, 0, 360 ) and the general solution e) 4sin + = 5, 0 and the general solution f) 4csc 3 5 + =, 0, 360 ) and the general solution
73 Level II: Solve by factoring Ex) Solve the following. a) cos x cos x= 0, 0 x 360 and the general solution b) sin x tan x = sin x, 0 x and the general solution c) cos csc cos 0 + =, ) 0, and the general solution
74 d) sin x sin x 1= 0, 0, 360 ) and the general solution e) sin x 7sin x 4 + =, ) 0, and the general solution f) csc 3csc 8 0 =, 0 360 and the general solution
75 Level III: These involve multiple angle equations Ex) 1 sin x =, cos3 =, 1 tan 1 0 x = Solve for x, or 3, or 1 x, etc, listing all principle solutions and general solution then give answers for just x or. Ex) Solve the following. a) cos + 1 = 0, 0 360 and the general solution
76 b) sin x 1 =, ) 0, and the general solution c) sin 1 3 x =, 0 x and the general solution d) sec3 =, 0,360 ) and the general solution
77 Solving Trigonometric Equations Algebraically Assignment: 1) Solve the following Level I equations. In each case, provide answers for the specified domain and provide the general solution. a) 3 sin x =, 0 x 360 b) tan x = 1, 0x c) cos = 0, x d) 3cot + 7 = 6, 0 x 360 e) 5sin x= 3sin x 1, 0, 360 ) f) 3sec 1 1 x =, 0, ) g) 5csc + 3 = 13, 0, ) h) 7cos 4 3 =, 0, 540 )
78 ) Solve the following Level II equations. In each case, provide answers for the specified domain and provide the general solution. a) =, 0, 360 ) sin x sin x 0 b) sin cos + cos = 0, 0, ) c) tan xsec x tan x = 0, 0x d) cos 3cos 1 0 + =, 0 360
79 e) tan x tan x 0 =, 0, ) f) =, 0, 360 ) sec sec 3 0 g) csc 3csc =, 0 360 h) =, 0x 6sin x 5sin x 1
80 3) Solve the following Level III equations. In each case, provide answers for the specified domain and provide the general solution. a) 3 sin x =, 0 x 360 b) cos3 +, 0 c) cot x 1= 0, 0, 360 ) d) sin3 cos3 + sin3 = 0, 0, ) e) 1 1 sin x sin x 1= 0, 0x
81 Trigonometric Identities: We already know some basic trigonometric identities: csc = tan = 1 sin sin cos sec = 1 cos 1 cos cot = = tan sin New identities: sin + cos = 1 tan + 1= sec 1+ cot = csc Proof:
8 Strategies for proving identities: Convert everything so it is written in terms of sine or cosine Simplify Try to make one side of the identity match the other ** Remember: We must treat each side of the identity as independent of one another. We can not treat these like an equation. Ex) Prove the following identities. a) sin x + cot xcos x = csc x
83 tan A b) sin 1+ tan A = Acos A c) 1 1 + = csc 1+ cos x 1 cos x x
84 d) cos x 1+ sin x = 1 sin x cos x e) Verify the identity sin x + cot xcos x = csc x for x = 6
85 Trigonometric Identities Assignment: 1) Determine the non-permissible values of x, in radians, for each expression. a) cos x sin x b) sin x tan x c) cot x 1 sin x d) tan x cos x + 1 ) Simplify each expression to a single trigonometric function sin x, cosx, tan x, cscx, secx, or cot x. a) sec xsin x b) sec xcot xsin x c) cos x cot x d) cos x tan x tan x sin x cos x e) csc xcot xsec xsin x f) 1 sin x
86 3) Verify that the equation sec x sin x tan x+ cot x = is true for x =. 4 4) Verify that the equation sin xcos x 1 cos x = 1+ cos x tan x is true for x = 30. 5) Compare y= sin x and y = 1 cos x by completing the following. a) Verify that sin x 1 cos = x is true for 60 x =, x = 150, and x = 180. b) Graph y= sin xand c) Determine whether y = 1 cos x in the same window. sin x 1 cos = x is an identity. Explain your answer.
87 6) Simplify ( sin x cos x) ( sin x cos x) + +. 7) Determine an expression for m that makes cos sin x x = m+ sin x an identity. 8) Prove the following identities and state any restrictions that may apply. a) 1 1 + = sec 1+ sin x 1 sin x x
88 b) sin x sin xcos x sin x = sin x sin cos sin c) = csc cot cos 1 d) cos + cos tan = sec
89 e) sin x cos x sin x+ cos x = sin x cos x f) 1 sin x 1+ sin = 1+ sin 3sin 1+ 3sin x x x x g) sin cos cot + = 1+ cos sin 1+ cos
90 h) cos x cos x + = cot x sec x 1 sec x+ 1 i) sin + cos cot = csc j) sin cos 1 cos = 1+ cos tan
91 Sum and Difference Identities: ( ) ( ) ( ) ( ) sin A+ B = sin Acos B + sin Bcos A sin A B = sin Acos B sin Bcos A cos A + B = cos Acos B sin Asin B cos A B = cos Acos B + sin Asin B tan A+ tan B tan( A+ B) = 1 tan Atan B tan A tan B tan( A B) = 1 + tan Atan B ( A) = A A cos( ) = cos sin sin sin cos A A A tan A tan( A) = cos( A) = cos A 1 1 tan A cos A( A) = 1 sin A Ex) Use the above identities to write the following as a single trigonometric function. a) sin 0 cos3 sin3 cos 0 b) cos15 cos30 sin15 sin30
9 Ex) Determine the exact value of sin 75 cos15 cos75 sin15 Ex) Use the sum and difference identities to determine the exact value of the following. a) cos15 b) sin 3
93 c) cos 15 sin 15 d) 5 tan 1 Prove the following trigonometric identities. a) cos( x + y)cos y + sin( x + y)sin y = cos x
94 b) 1+ cot xtan y= sin( x+ y) sin xcos y c) sin( )sin( ) cos cos x + y x y = y x
95 Sum and Difference Identities Assignment: 1) Write each expression as a single trigonometric function. a) cos43 cos7 sin 43 sin 7 b) sin15 cos 0 + cos15 sin 0 c) cos 19 sin 19 d) 3 5 3 5 sin cos cos sin 4 4 e) 8sin cos 4 4 tan 76 f) 1 tan 76 g) 1 cos 1 h) ( ) 6cos 4 6sin 4 tan 48 ) Simplify and then give the exact value for each expression. a) cos40 cos0 sin 40 sin 0 b) sin 0 cos5 + sin 5 cos0 c) cos sin d) cos cos sin sin 6 6 3 3
96 3) Simplify cos( 90 x) using a difference identity. 4) Determine the exact value for each of the following. a) cos75 b) tan165 c) 7 sin 1 d) sec195
97 5) Angle is in quadrant II and the following. 5 sin =. Determine an exact value for each of 13 a) cos b) sin 6) If the point (, 5 ) lies on the terminal arm of angle in standard position, determine the value cos( + ). 7) What value of k makes the equation sin5xcos x + cos5xsin x = sin kxcos kx true?
98 8) If A and B are both in quadrant I, and each of the following. 4 sin A = and 5 1 cos B =, evaluate 13 a) cos( A B) b) sin ( A+ B) 9) Prove the following identities and state any restrictions that may apply. a) 4 4 cos + sin = cos x x x b) 1 cosx = sin x
99 c) 4 8sin 4 = sin cos tan d) csc x cscx cos x = sin + tan sin e) = 1+ cos cos
100 f) sin x cosx + = csc x cos x sin x g) sin( 90 + ) = sin( 90 ) h) sin4x sinx tan x = cos4x+ cosx
Solving Trigonometric Equations Algebraically: (Part ): Type IV: These equations first require you to make a substitution using a trigonometric identity so that the equation becomes a Type I, II, or III question. Ex) Solve the following. a) cos x 1 cos x 0 + =, 0,360 ) and the general solution 101 b) 1 cos x 3sin x =, 0 x and the general solution
10 c) sinx cos x =, ) 0, and the general solution d) sin x= 7 3csc x, 0 360 and the general solution
103 Solving Trigonometric Equations Algebraically Part II Assignment: Solve the following equations. In each case, provide answers for the specified domain and provide the general solution. 1) sin x sin x= 0, 0 x 360 ) cos x sin x =, 0, ) 3) cos cos = 0, 0 4) tan cos sin 1= 0, 0, 360 )
104 5) cos x 3sin x =, 0, ) 6) 3csc sin =, 0 x 360 7) = +, 0 sin x cos x 1 8) sin x = cos x cosx, 0, 360 )
105 Answers: Trigonometry Fundamentals Assignment: 1. a) 5 7, 0.5 b), 5.4 c), 0.37 d), 1.57 6 3 60 e) 5 3, 13.09 f),.36 6 4. a) 60 b) 5 c) 10 d) 157.56 e) 756 f) 57.30 3. a) 4 b) 1 c) d) 4 e) 3 f) 4. Possible answers could be: a) 88, 43 b) 3 5, 7 7 c) 565, 155 d) 3.37,.9 e) 00, 160 f) 4, 3 5. a) =.5 or = 43 b) r = 3.8 cm c) a = 17.10 m d) a = 10.98ft 6. a) 6 or 3.77 or 16 b) 47.1 m 5 7. Revolutions Degrees Radians 1 rev 360 3 4 rev 70 3 5 1 rev 150 56 0.3 rev 97.4 1.7 1 9 rev 40 9 0.7 rev 5 7 5 3.5 rev 1170 13 3 18 rev 460 3 9 3 16 rev 67.5 3 8 3
106 8. a) b) sin 45 =, cos45 =, tan 45 = 1, csc45 =, sec45 =, cot 45 = 1 sin 40 3 1 =, cos 40 =, tan 40 = 3, csc 40 =, 3 sec40 =, cot 40 = 1 3 5 1 5 3 5 1 5 5 c) sin =, cos =, tan =, csc =, sec =, 6 6 6 3 6 6 3 5 cot = 3 6 3 3 3 d) sin = 1, cos = 0, tan undefined 3 =, csc = 1, 3 sec = undefined, cot 45 = 1 9. a) 1 & 4 b) & 4 c) 1 & d) e) f) 1 10. a) sin 50 = sin 70 b) tan 90 = tan70 c) sec135 = sec45 cos4 cos 4 csc3 csc 3 cot 4.95 = cot 4.95 d) = ( ) e) = ( ) f) ( ) 11. a) 1 b) c) 1 d) 7 e) 1 f) 3 1. a) 7 11 3 5 =, b) =,, 6 6 4 4 4 c) = 60, 60 d) = 360, 180, 0, 180 13. a) 4 3 5 5 4 cos =, tan =, csc =, sec =, cot = 5 4 3 4 3 b) 1 1 3 sin =, tan =, csc = 3, sec =, cot = 3 c) sin = 3 13 13 3, cos =, csc =, sec =, cot = 13 13 3 13 3 39 d) sin =, cos =, tan =, 4 4 3 3 4 3 4 5 14. sin =, cos =, tan =, csc =, 5 5 3 4 4 3 csc =, cot = 13 39 5 3 sec =, cot = 3 4
107 15. 16. sin =, 9 9 sec =, 5 1 sin =, 13 5 cot = 1 5 9 cos =, tan =, csc =, 9 5 5 cot = 5 1 cos =, tan =, 13 5 13 13 csc =, sec =, 1 5 Unit Circle Assignment: 1. a) No b) Yes c) No d) Yes e) Yes f) Yes. a) 1 15 5 7 15, b), c), 4 4 3 3 8 8 d) 6 5 1 1 5, e), f), 7 7 3 3 13 13 3. a) ( 1, 0) b) ( 0, 1) c) 1 3, d), e) ( 0, 1 ) f) 3 1, 4. a) 3 5 5 = b) = c) = d) = 4 3 6 e) = 0 f) 7 = 6 5. 150 or 5 7 & 10 or 6 6 6. a) 4 4 5 y = b) tan = c) csc = 5 3 4 Graphing Sine & Cosine Functions Assignment: 1. a) Amplitude =, Period = 60 or b) 3 1 Amplitude =, 3
108 Period = 1080 or 6 c) Amplitude = 16, 300 5 d) Amplitude = 15, Period = or 3 y y 8, y R y 5 y 1, y R. a) b) c) y.5 y 5.5, y R d) 3. a) i b) iv c) iii d) ii 4. y = 15cos3 + 9 9 5. a) iv b) ii c) iii d) i ( ) 6. y ( ) 7. a) = 16sin 9 5 + 3 Period = 40 or 1 17 y y, yr 1 1 4 3 b)
109 c) d) 8. 1 9. 0.9 10. a) 1 1 y = sin( 90 ) +, y = cos( 180 ) + b) 7 1 7 1 5 y = sin + + 1 4 6, y = cos 1 4 + 6 c) 3 4 3 3 4 5 3 y = sin + 3 4, y = cos + 3 8 d) 5 1 5 1 y = sin, y = cos 4 6 4 4 3 4 11. a) 10 b) 3 15 c) d) 4 4
110 Applications of Sine & Cosine Functions Assignment: 1. a) Maximum = 78 decibels, Minimum = 4 decibels b) 30 sec. c) x : 0, 9, 10 y : 0, 80, 10 (shows 3 cycles) d).8 sec. e) 19.7 sec. & 5.3 sec. f) 81%. a) 369 km North b) 3765 km North c) 35.3 min. 3. a) Maximum = 7.5 m, Minimum = 7.5 m b) 9.5 min. c) 5.9 m d) i) h 6 m ii) 3.8 min. iii) 1.9 min. 4. a) 1 hours b) 7.9 m 5. a) b) Yes it will run dry in July of 004. 6. a) 11 m b) h= 9sin t+ 11 16 c) 4.6 m d) 8.3 sec. 7. a) b) h( t) = 15cos ( t 5) + 16 5 e) 6 sec. c) 11.4 m d).1 sec.
111 8. a) b) 1.75 m c) h( t) = 1.5sint + 1.75 d).8 m e) 1. sec. Graphs of Other Trigonometric Functions Assignment: 1. Period 360 =, Domain: 45 n90, n I, R Range: y 4 y 10, y R. Period 540 3. Period 540 +, =, Domain: 70 n540, n I, R =, Domain: 40 + n70, ni, R, Range: y y 7, y 17, y R 4. n60, ni, R +, Range: y y R Solving Trigonometric Equations Graphically Assignment: 1. a) b) 4 c) 3 d). a) 5 = + n, n I b) = + n, n I 3 3 3. a) = 1.35, 4.49 b) = 1.14,.00 c) = 1.0, 5.08 d) = 3.83, 5.59 4. a) = 0 b) x = 315, 5, 45, 135 c) 5 11 =,, 6 6 6 d) 5 5 7 17 x =,,, 6 6 6 6 5. sin = 0 has solutions of 0,, and, but none of these are included in the,. interval ( )
11 Solving Trigonometric Equations Algebraically: 1. a) 60 + n360 x = 60, 10 ; x = 10 + n360, nr b) 3 7 3 x =, ; x = + n, n R 4 4 4 c) 7, + n 4 =, ; =, nr 4 4 4 7 + n 4 d) = 10, 300 ; = 10 + n180, n R e) 10 + n360 x = 10, 330 ; x =, nr 330 + n360 f) + n 11 6 x =, ; x =, nr 6 6 11 + n 6 g) 3 5 7,,, = ; = + n, n R 4 4 4 4 4 h) = 0, 360 ; = n360, n R. a) n180 x = 0, 90, 180 ; x = 90 + n360, nr b) 7 3 11 =,,, + n 3 ; =, nr 6 6 3 + n c) x = 0, ; x = n, n R n360 d) = 0, 60, 300 ; = 60 + n360, n R 300 + n360
113 e) 1.107 + n 3 7 x = 1.107,, 4.5, ; x = 3, nr 4 4 + n 4 f) 70.5 + n360 = 70.5, 180, 89.5 ; = 180 + n360, n R 89.5 + n360 g) 30 + n360 = 30, 150 ; = 150 + n360, nr 0.34 + n 0.5 + n h) x = 0.34, 0.5,.6,.80 ; x =.6 + n.80 + n 3) a) 30 + n x = 30, 60, 10, 40 ; x =, nr 60 + n b) 5 11 13 19 7,,,,, + n 4 3 = ; =, nr 4 1 1 1 1 4 5 + n 1 3 c) x =.5, 11.5, 0.5, 9.5 x =.5 + n90, n R d) 4 5 0,,,,, = ; = n, n R 3 3 3 3 3 4 e) x = ; x = + n, n R 3 Trigonometric Identities Assignment: 1. a) x n b) x n c) x n, x + n d) x + n, + n. a) tan x b) sin x c) sin x d) cot x e) cscx f) secx
114 Sum & Difference Identities Assignment: 1. a) cos70 b) sin35 c) cos38 d) sin 4 e) 4sin f) tan15 g) cos 6 h) 6sin 48. a) 1 b) 3. sin x 4. a) 6 4 5. a) 119 169 b) 6. 9 7. k = 3 8. a) 56 65 b) c) 1 d) b) 3 c) 10 165 63 65 1 6 + 4 d) 4 6 + Solving Trigonometric Equations Algebraically Part II Assignment 1. n180 x = 0, 60, 10, 180, x = 60 + n360, n I 10 + n360. 5 3 x =,, ; x = + n, n I 6 6 6 3 3. 4 = 0,, ; = n, n I 3 3 3 4. = 90, 70 ; = 90 + n180, n I
115 5. 7 + n 6 7 3 11 3 x =,, ; x = + n, n I 6 6 11 + n 6 6. = 90 ; = 90 + n360 7. 3 =, ; = + n, n I 8. 30 + n10 x = 30, 90, 150, 70 ; x = 90 + n360, ni