Fall 2015 Trigonometry: Week 7 Today s Topics/Activities: 1. More Practice Solving Equations 2. More Practice Graphing and Unsolving (by framing) 3. Ungraphing From Data Points 4. Ungraphing From Stories 5. (Optional): Finding Trig Function Values from a Calculator Unit Circle and Using Laps 6. The Graphs of the Reciprocal Trig Functions: cot(t), sec(t), csc(t) 7. Finding Rough-And-Dirty Inverse Trig Function values: sin 1 (y), cos 1 (x), tan 1 (m) 8. (Optional): The Graphs of the Inverse Trig Functions
I. More practice solving equations: (a) 5 cos(2(t + 1)) 3 = 0 (b) 2 tan(0.3(t + 2)) 40 = 0 (c) 4 csc(4t) = 0.1 (d) 3 sec(0.1(t 5)) = 7 (e) 4 cot(7t + 1) = 9
II. More practice graphing and ungraphing (by framing): (a) without your calculator sketch the graph of y = 4 + 2 sin you used each of the coefficients to obtain part of the frame: ( 1 3 (t + 4) ). Show how (b) Find an equation of the graph below. Explain how you used various aspects of the graph to come up with the equation:
III. Ungraphing Data: example: The average daily temperature T (in degrees Fahrenheit) is given in the table below. Time t is measured in months, with t = 0 representing January 1. Find a trigonometric model (sometimes called a sinusoidal model) that gives T as a function of t. That is, find a trig equation whose values are close to those given below: t 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 T -10.1-3.6 11.0 30.7 48.6 59.8 62.5 56.8 45.5 25.1 2.7-6.5 Procedure/hints: Using your calculator or graph paper, plot the points, then connect them with a triglooking graph Use the method of ungraphing trig functions to come up with the graph solution: Here are the points all graphed, with the frame and middle line drawn in: We see that this is the graph of an upside down cosine graph (and we ll have to remember that as we put the equation together). Remember that we re looking for an equation of the form y = A + B cos(c(t D)), where 62.5 + 10.1 A is the height of the middle line = = 27.1 2 B is the distance between the middle line and the top = 62.5 27.1 = 35.4 D is the location of the left side of the frame = 0 C is found by taking the width of the frame, in this case, 12, and dividing into 2π, and getting 2π 12
We put this all together, remembering that we re seeing an upside down cosine graph, as ( ) 2π y = 27.1 35.4 cos 12 (t 0). Your turn: (Taken from Essentials of Trigonometry, 4th edition, Karl J. Smith): The fly wheel on a lawn mower has 16 evenly spaced cooling fins. If we rotate the engine through two complete revolutions, we can generate 32 data points, with the first coordinate representing time and the second representing the depth of the piston. Determine an equation of the curve of the generated data: x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 y 0 0.2 0.8 1.7 2.5 3.4 4.0 4.4 4.5 4.4 4.0 3.5 2.2 1.7 0.9 0.2 x 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 y 0 0.2 0.8 1.7 2.5 3.4 4.0 4.4 4.5 4.4 4.0 3.5 2.2 1.7 0.9 0.2 0 IV. From Stories: example: A Ferris wheel with diameter of 50 feet is rotating at a rate of 3 revolutions per minute. When t = 0, a chair starts at the lowest point on the wheel, which is 5 feet above the ground. Find a sinusoidal model for the height h (in feet) of the chair as a function of the time t (in seconds). Procedure/hints: Make a table of easily-determined values of times t and heights h. Since we re following an object in a circular motion (in this case, a seat on a ferris wheel), there are (at least) 4 very easy times/heights to find: t h Once you have you very simple table, plot the points, connect the dots in a sinusoidal manner, then sketch the frame: Now ungraph-by-framing to come up with your equation:
your turn: (Source: S.S. Beaver: The Ship That Saved the West): The paddle wheel of the S.S. Beaver was 13 feet in diameter and revolved 30 times per minute when moving at top speed. Using this speed and starting from a point at the very top of the wheel, find a model for the height h (in feet) of the end of a paddle relative to the water s surface as a function of the time t (in minutes). You will need to know that the paddle is 2 feet below the water s surface at its lowest point). V. How to use your graphing calculator s unit circle and the notion of laps to get good cos(t) and sin(t) approximations. example: Use your graphing calculator s unit circle to approximate sin(2000). First figure out how many whole laps we would have to go. We do this by dividing 2000 by 2π and getting 2000 2π = 318.3098861, to 7 decimal places. We see that we went 318 whole laps (which doesn t affect at all where we end up on the circle), and that we had to go another 0.3098861 of a lap (a little less than a third). To find the distance (and not fraction of a lap) that we could go and still end up at that point, we need to multiply to 7 decimal places. 318.3098861 2π = 1.9470718, If we now want to use our unit circle, we trace to t = 1.9 on the unit circle, and read off the y-coordinate as y = sin(t) = 0.9463001 VI. The other trig graphs: There are three other trig functions whose basic graphs we haven t explored. These are each reciprocal graphs of the ones we have. To graph each, we start with the function we know, and apply the following facts: When a function s graph has a y-value of ±1, the reciprocal graph has a y-value of ±1 When a function s graph has a y-value of 0, the reciprocal graph (usually) has an asymptote there When a function s graph has small y-values, the reciprocal graph has a large y-value there When a function s graph has large y-values, the reciprocal graph has small y-values there Although you will not be required to derive each of these graphs, you will be required to sketch accurate graphs of the reciprocal trig functions from good graphs of the basic three trig functions that you will be drawing.
csc(t) = 1/ sin(t) sec(t) = 1/ cos(t) cot(t) = 1/ tan(t)
sin(t) cos(t) tan(t)
csc(t) = 1/ sin(t) sec(t) = 1/ cos(t) cot(t)t) = 1/ tan(t)
VII. Inverse trig functions: A. Review: For values of sin 1 (y) = arcsin(y), cos 1 (x) = arccos(x), and tan 1 (m) = arctan(m), we see that the distance t that each of these outputs is (a) the smallest distance to a correct point and (b) positive if possible. Remember that each of these is a function and only outputs one number!!! i. sin 1 ( 0.9) ii. cos 1 ( 0.9) iii. tan 1 ( 0.9) your turn: Using just the unit circle, find rough-and-dirty values of each of the following: i. arcsin(0.7) ii. arccos(0.1) iii. arctan(5) iv. arcsin(5)
B. Simplifying expressions that involve both trig and inverse trig functions of triangle angles. This skill will be needed when solving more complicated equations, but where we want the answers to be as simple as possible: example: Simplify the expression csc (cos 1 ( 2 7 As with all simplification of algebraic expressions, we will try to start from the inside and work our way out. The very insidiest part of this is 2 7 which we can t simplify at all! So, we move just a little more outside to ( cos 1 2 ) 7 which we ll need both the circle and the triangle to understand. Recall that we plug x-values into cos 1, and the answers are t values (distances around the circle). Since 2 0.3, we can draw a circle along with the line x = 0.3 to get an idea of where our 7 angle sits: )) : Since the inverse trig function values are always the closest distance, positive if possible, the point we re interested in is the one labeled A. Since we re being asked to find the exact cosecant, we need to draw and label the corresponding triangle, which we ve done before: Finally, we re being asked for the cosecant of the central angle (the one at the origin), and this is csc(t) = 1 sin(t) = 1 = 7. 45 45 7
your turn: i. cos(tan 1 (4)) ii. sec(sin 1 (0.8)) iii. cot(cos 1 (0.2)) iv. cot(sin 1 (x))
C. The graphs of the inverse trig functions. (This will not be required material on any test, but will show up as extra credit occasionally): Recall that the inverse trig functions are obtained by switching the input numbers with the output numbers. Because of this, we can obtain the graphs of the inverse trig functions by switching all of the x and y-values of the originals, then erasing enough so that what s left is a function. So, for each of the following, cross off enough of the graph so that what is left passes the vertical line test and so that we leave the graph that is closest to the origin untouched (then check with your calculator to double-check we created the right graph): a. b. c.
VIII. Homework: 1. Solve each of the following equations, finding all solutions, using your calculator to help you give very precise answers: (a) 5 12 cos(3(t + 1)) = 8 (b) 3 + 4 sin(4.1(t 3)) = 1.2 (c) 2 + 7 tan(π(t + 1)) = 40 2. Find two equations for each of the graphs below, one involving sine, the other involving cosine:
3. Find a sinusoidal model (so, an equation of a general sine or a general cosine function) for each of the following: (a) In front of the Antique Sewing Machine Museum in Arlington, Texas, is the largest sewing machine in the world. The flywheel, which turns as the machine sews, is 5 feet in diameter. The wheel makes a complete revolution every 2 seconds and the handle starts at its minimum height of 4 feet above the ground. Find a model for the height h (in feet) of the handle on the flywheel as a function of time t (in seconds). (b) For any given day, the number of degrees that the average temperature is below 65 F is called the degree-days for that day. This figure is used to calculate how much is spent on heating. The table below gives the total number T of degree-days for each month t in Dubuque, Iowa with t = 1 representing January. Find a model for the data: t 1 2 3 4 5 6 7 8 9 10 11 12 T 1420 1204 1026 546 260 78 12 31 156 450 906 1287 4. Using both the unit circle and a reference triangle, find the exact values of each of the following: ( ( )) 7 (a) tan sin 1 25 ( ) (b) sec tan 1 (11) 5. Be prepared to graph all six trig functions on the same axes (the first time will be for an extra credit quiz) when you come back, as well as be able to graph each of our three basic inverse trig graphs from the original trig graphs 6. Estimate each of the following using our unit circle with t-step set at 0.05, by adding or subtracting an appropriate number of laps, then using your calculator s unit circle and tracing. Remember to show all of your work: (a) sin(450) (b) cos( 2000)
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