(1) Trigonometry 7

(1) Trigonometry 7 - Pre – Calculus Math 40S Explained...

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Unformatted text preview: Pre – Calculus Math 40S: Explained! www.math40s.com 67 TRIGONOMETRY LESSON SEVEN Part I Graphing Radian Functions In application questions, when your x-axis is time, distance, or some other unit, you must do your trig graph in radian mode. Basically, whenever you see integers on the x-axis, (rather than degrees or radian fractions) you need to be in radian mode. In the graph on the right, notice how the variable is time instead of θ. This is a clue you need radian mode. y =1.23cos 2π t +1.85 12.1 Example 1: Given the equation: y =13.2cos 2π (t -101)+6.5 find 342 appropriate units for your window and graph with the TI-83. 1) First you must figure out the period so you know how long a cycle is. 2π 2π 342 P er iod = = = 2π × = 342 2π b 2π 342 The length of one complete cosine cycle is 342 units. Also, since the pattern starts at 101 due to the phase shift, we want to see all the way to 443 (phase shift + period) for the complete picture. 2) Next you need to know where the minimum and maximum values are using the following formulas: Minimum = d — a Maximum = d + a Minimum = 6.5 — 13.2 Maximum = 6.5 + 13.2 Minimum = -6.7 Maximum = 19.7 3) Now choose a scale. The best way to do this is by picking numbers large enough that you won’t have too many ticks on either axis. For the x-axis, we’re going from 0 to about 450, so use a scale of 100. For the y-axis, since we want to see from -7 to +20, a scaling of 5 would be good. Of course, these are just guidelines and you could use several different scales and still obtain a good graph. 5) Now draw the graph: 4) Use the following y window settings: 20 Xmin: 0 Xmax: 443 Xscl: 20 Ymin: -6.7 Ymax:19.7 Yscl: 5 15 10 5 t 100 200 300 400 -5 Pre – Calculus Math 40S: Explained! www.math40s.com 68 TRIGONOMETRY LESSON SEVEN Part I Graphing Radian Functions Questions: For each of the following equations, find appropriate window settings. Then draw the graph using your TI-83. 1. π y =17.2cos (t -7)+ 0.5 3 y = 2.2sin0.123π (t +1.7)+15.2 2. 2 0 2 0 1 5 1. 75 1 0 1 5 5 1. 25 1 0 2 4 6 8 1 0 1 2 5 75 . -0 1 5 -5 1 25 . -0 2 3. 2 y = 20.1sin 2π (t - 265)+ 6.2 300 4. 4 6 8 1 0 1 2 1 4 y = -3.2cos0.18π t +17 2 0 3 0 1. 75 1 5 2 0 1. 25 1 0 1 0 75 . 10 0 20 0 30 0 40 0 50 0 60 0 5 25 . -0 1 2 4 6 8 1 0 1 2 Pre – Calculus Math 40S: Explained! www.math40s.com 69 TRIGONOMETRY LESSON SEVEN Part I Answers: 1. 1) Find the period: Period = Graphing Radian Functions 2π 2π 3 = = 2π × = 6 π b π 3 Since the cosine pattern starts at 7 due to the phase shift, and the period is 6, we want to see up to 13 on the x-axis. 2) Find the minimum and maximum: Minimum = d — a = 0.5 — 17.2 = -16.7 Maximum = d + a = 0.5 + 17.2 = 17.7 5) Draw the graph. 3) Choose a scaling, 2 for the x-axis and 5 for the y-axis will work fine. 4) Use the following window settings: Xmin: 0 Xmax: 13 Xscl: 2 Ymin: -16.7 Ymax:17.7 Yscl: 5 2. 1) Find the period: Period = 2π 2π = = 16.23 b 0.123π Since the sine pattern starts at -1.7 due to the phase shift, and the period is 16.23, we want to see at least up to 14.53 on the x-axis for the complete picture. 2) Find the minimum and maximum: Minimum = d — a = 15.2 — 2.2 = 13 Maximum = d + a = 15.2 + 2.2 = 17.4 3) Choose a scaling: 2 for the x-axis and 5 for the y-axis will work fine. 4) Use the following window settings Xmin: 0 Xmax: 14.5 Xscl: 2 Ymin: 0 Ymax:17.4 Yscl: 2.5 We have two options for where to set the Ymin value. If we use 13, we’ll get a big display of the graph, but lose the frame of reference with the origin. If we use 0, we’ll keep our frame of reference (which can be very useful in applications), but have a small display. Pre – Calculus Math 40S: Explained! www.math40s.com 70 TRIGONOMETRY LESSON SEVEN Part I Graphing Radian Functions Answers: 3. 2π 2π 300 = = 2π × = 300 2π b 2π 300 Since the sine pattern starts at 265 and the period is 300, we should extend our x-axis to 565. 1) Find the period: Period = 2) Find the minimum and maximum: Minimum = d – a = 6.2 – 20.1 = -13.9 Maximum = d + a = 6.2 + 20.1 = 26.3 5) Graph the equation 3) Choose a scaling: 100 for the x-axis, and 10 for the y-axis. 4) Use the following window settings Xmin: 0 Xmax: 565 Xscl: 100 Ymin: -14 Ymax:27 Yscl: 10 4. 1) Find the period: Period = 2π 2π = = 11.11 b 0.18π 2) Find the minimum and maximum: Minimum = d — a = 17 — 3.2 = 13.8 Maximum = d + a = 17 + 3.2 = 20.2 5) Graph the equation 3) Choose a scaling, 2 for the x-axis and 5 for the y-axis. 4) Use the following window settings Xmin: 0 Xmax: 11 Xscl: 2 Ymin: 0 Ymax:27 Yscl: 5 Pre – Calculus Math 40S: Explained! www.math40s.com 71 TRIGONOMETRY LESSON SEVEN Part II Solving Radian Equations Example 1: The height of an object is given by the equation: h(t) = 2.2sin0.123π (t +1.7)+15.2 Find the height after 2.4 seconds. Alternatively, We may evaluate a point on a trig function by using the TI-83. h(2.4) = 2.2sin0.123π (2.4 +1.7)+15.2 (Radian Mode) h(2.4) =17.4 Questions: Evaluate the following functions for the value indicated. we could draw the graph and use: 2nd Trace Value x = 2.4 1) Evaluate h(t) =13.2cos 2π (t -101)+6.5 when t = 105. 342 2) Evaluate h(t) = 20.1sin 2π (t - 265)+ 6.2 when t = 296 300 3) Evaluate h(t) =18.5cos 2π (t - 28)+ 4.5 when t = 45 365 Answers: π 4) Evaluate h(t) =12sin (t - 3)- 3 when t = 4 2 1) 19.7 2) 18.4 3) 22.2 4) 9 Pre – Calculus Math 40S: Explained! www.math40s.com 72 TRIGONOMETRY LESSON SEVEN Part II Solving Radian Equations Sometimes the y-variable is given, and we need to find the x-variable. In this case, we must graph both the left side and right side of the equation, then find the intersection point. Example 1: The height of an object is given by the equation: h(t) = 2.2sin0.123π (t +1.7) +15.2 Find the time when the object first reaches a height of 17. In your graphing calculator, graph: y1 = 2.2sin0.123π (t +1.7)+15.2 y2 = 17 Now find the first intersection point. TI-83: 2nd Trace Intersect The x-value of the first intersection point will be the time the object first reaches a height of 17. Questions: Find the time when each of the following heights are reached. Don’t forget to set your window properly! 1) Find the time when a height of 12 is first reached in the equation: 2π h(t) = 13.2cos (t -101) + 6.5 342 2) Find the time when a height of 23 is first reached in the equation: 2π h(t) = 20.1sin (t - 265) + 6.2 300 3) Find the time when a height of 12 is first reached in the equation: 2π h(t) = 18.5cos (t - 28) + 4.5 365 4) Find the time when a height of 0 is first reached in the equation: h(t) = 12sin π 2 (t - 3) - 3 Pre – Calculus Math 40S: Explained! www.math40s.com 73 TRIGONOMETRY LESSON SEVEN Part II Solving Radian Equations Answers: 1) 2) t = 38.9 s 3) t = 12.3 s 4) t = 95 s t = 0.84 s Pre – Calculus Math 40S: Explained! www.math40s.com 74 ...
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This note was uploaded on 02/09/2011 for the course 168 comm 168 taught by Professor Kiscabean during the Winter '10 term at UCLA.

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