Exploration 6-1a: Sine and Cosine Graphs, Manually

Transcription

1 Group Members: Exploration 6-1a: Sine and Cosine Graphs, Manuall Objective: Find the shape of sine and cosine graphs b plotting them on graph paper Explorations Exploration Masters Precalculus with Trigonometr: Instructor s Resource Book 2012 Ke Curriculum Press Precalculus with Trigonometr Course Sampler 133

2 Group Members: Exploration 6-2a: Transformed Sinusoid Graphs Objective: Given the equation for a transformed sinusoid, sketch the graph, and vice versa. Explorations Exploration Masters Precalculus with Trigonometr: Instructor s Resource Book 2012 Ke Curriculum Press 134 Precalculus with Trigonometr Course Sampler

3 Group Members: Exploration 6-3a: Tangent and Secant Graphs Objective: Discover what the tangent and secant function graphs look like and how the relate to sine and cosine Explorations Exploration Masters Precalculus with Trigonometr: Instructor s Resource Book 2012 Ke Curriculum Press Precalculus with Trigonometr Course Sampler 135

4 Group Members: Exploration 6-3b: Transformed Tangent and Secant Graphs Objective: Sketch transformed tangent, cotangent, secant, and cosecant graphs, and find equations from given graphs. Explorations Precalculus with Trigonometr: Instructor s Resource Book Exploration Masters Ke Curriculum Press 136 Precalculus with Trigonometr Course Sampler

5 Group Members: Exploration 6-7a: Oil Well Problem Objective: Use sinusoids to predict events in the real world Fence Inaccessible land x = 700 ft Available land x = 2000 ft = 2500 ft Top surface Explorations 182 Exploration Masters Precalculus with Trigonometr: Instructor s Resource Book 2012 Ke Curriculum Press Precalculus with Trigonometr Course Sampler 137

6 Group Members: Exploration 6-8b: Motorccle Problem Objective: Find angular and linear velocities of connected rotating objects. Explorations Precalculus with Trigonometr: Instructor s Resource Book Exploration Masters Ke Curriculum Press 138 Precalculus with Trigonometr Course Sampler

7 Group Members: CAS Activit 6-4a: Inverse Trigonometric Functions Objective: Prove co t 1 x = ta n 1 1 x and its parallel secant and cosecant forms. Explain wh onl three inverse trigonometric functions are required. Technolog Activities 348 CAS Activities Precalculus with Trigonometr: Instructor s Resource Book 2012 Ke Curriculum Press Precalculus with Trigonometr Course Sampler 139

8 Group Members: CAS Activit 6-7a: Epicenter of an Earthquake Objective: Discover the minimum number of points required to definitivel locate the source of an earthquake. Zeros Zeros(dist(0, 0, x, ) dist(432, 7, x, ) , ) Technolog Activities Zeros Define dist(a,b,c,d) = Precalculus with Trigonometr: Instructor s Resource Book CAS Activities Ke Curriculum Press 140 Precalculus with Trigonometr Course Sampler

9 Transformations of Circular Functions In this activit ou will use a point on the unit circle to construct dilated images of circular functions. A F C D 2 SKETCH AND INVESTIGATE 1. Open Circular Transforms.gsp. The sketch contains a parameter k that currentl equals 2. Use the Calculator to multipl k b the angle measure of D C. To mark the calculation as the angle of rotation, select it and choose TransformMark Angle. To turn on tracing, select the point and choose DisplaTrace Point. Choose DisplaErase Traces to erase existing traces. 2. Mark point A as the center of rotation using TransformMark Center. Similarl, mark the calculation from step 1 as the angle of rotation. 3. Rotate point D b selecting it and choosing TransformRotate. Label the rotated point F, and construct segment AF. Q1 Press the Animate Point C button. What is the relation of DAC to DAF? Q2 For ever complete trip that point C makes around the circle, how man times does point F travel around the circle? Q3 Double-click parameter k, and change its value to 3. Answer Q1 and Q2 again for this new value. 4. Press the Show Point E button. This point, which ou built in the activit Trigonometr Tracers, traces out sin(md C ). Press the Animate Point C button to watch point E in action. Q4 You re about to create the graph of sin(kmd C. Before ou do, make a prediction: Based on our answers to Q2 and Q3, what do ou predict the graph will look like? 5. Measure F b selecting point F and choosing MeasureOrdinates (). 6. Plot the point md C, F b selecting in order md C and F, and then choosing GraphPlot as (x, ). 7. Label the plotted point G, and turn on tracing for it. Q5 Animate C, and observe the trace of G. Is our prediction about the graph of sin(kmd C correct? 8. Change the value of parameter k to draw new sine curves. Q6 B taking new measurements, create the graphs of cos(kmd C and tan(kmd C. Describe the appearance of each of these functions. Technolog Activities Precalculus with Trigonometr: Instructor s Resource Book The Geometer's Sketchpad Activities Ke Curriculum Press Precalculus with Trigonometr Course Sampler 141

10 Transformations of Circular Functions ACTIVITY NOTES SKETCH AND INVESTIGATE Q1 DAF is twice as large as DAC. Q2 Point F travels twice around the circle for ever revolution of point C. Q3 When k 3, DAF is three times as large as DAC, and F travels three times around the circle for ever revolution of C. Q4 Predictions will var. The important thing is that students make a prediction. Q5 The graph is a sine graph compressed in the x direction. It has an amplitude of 1 and a period of 2/3 so that it shows 3 complete ccles between 0 and 2. Technolog Activities EXTENSION 8. When ou change k, the period becomes 2/k, and the graph shows k complete ccles between 0 and 2. Q6 The graphs of these functions resemble the graphs produced in the Trigonometr Tracers activit, but (like the sine plot) compressed in the x direction so that the show k ccles between 0 and 2. You could challenge students to find a wa to modif the construction to produce vertical dilation in the resulting graph. One method would be to put a point on segment AF and plot the point s -coordinate as a function of the angle. If segment AF is constructed as a ra, it s possible to produce both compression and stretching. Alternativel, ou could dilate point F toward or awa from center point A. PRESENT To present this activit to the whole class, use Circular Transforms Present.gsp 418 The Geometer's Sketchpad Activities Precalculus with Trigonometr: Instructor s Resource Book 2012 Ke Curriculum Press 142 Precalculus with Trigonometr Course Sampler

11 Group Members: Problem Set 6-2/Pages Blackline Masters Precalculus with Trigonometr: Instructor s Resource Book Blackline Masters Ke Curriculum Press Precalculus with Trigonometr Course Sampler 143

12 Group Members: Problem Set 6-4/Pages Blackline Masters Precalculus with Trigonometr: Instructor s Resource Book Blackline Masters Ke Curriculum Press 144 Precalculus with Trigonometr Course Sampler

13 Test 15, Sections 6-1 to 6-3 Objective: Draw graphs of sinusoids and of tangent and secant functions. Form A Part 1: No calculators allowed (1 9) Assessment Resources Precalculus with Trigonometr: Assessment Resources Section, Chapter, and Cumulative Tests Ke Curriculum Press Precalculus with Trigonometr Course Sampler 145

14 Test 15, Sections 6-1 to 6-3 continued Form A Assessment Resources Part 2: Graphing calculators allowed (10 24) Rotation Ferris wheel Ground Seat 62 Section, Chapter, and Cumulative Tests Precalculus with Trigonometr: Assessment Resources 2012 Ke Curriculum Press 146 Precalculus with Trigonometr Course Sampler

15 Test 15, Sections 6-1 to 6-3 Objective: Draw graphs of sinusoids and of tangent and secant functions. Form B Part 1: No calculators allowed (1 9) a b A B Assessment Resources Precalculus with Trigonometr: Assessment Resources Section, Chapter, and Cumulative Tests Ke Curriculum Press Precalculus with Trigonometr Course Sampler 147

16 Test 15, Sections 6-1 to 6-3 continued Form B Assessment Resources Part 2: Graphing calculators allowed (10 24) Rotation Ferris wheel Seat Ground 64 Section, Chapter, and Cumulative Tests Precalculus with Trigonometr: Assessment Resources 2012 Ke Curriculum Press 148 Precalculus with Trigonometr Course Sampler

17 Chapter 6 Applications of Trigonometric and Circular Functions Problem Set Problem Set Solutions Manual Precalculus with Trigonometr: Solutions Manual Problem Set Ke Curriculum Press Precalculus with Trigonometr Course Sampler 149

18 Solutions Manual Problem Set 6-2 Precalculus with Trigonometr: Solutions Manual 2012 Ke Curriculum Press 150 Precalculus with Trigonometr Course Sampler

19 Dnamic Precalculus Exploration Experience the online version of this exploration at Variation of Tangent and Secant The sketch below will help ou understand how the functions tangent and secant var as their arguments var. Sketch This sketch shows a unit circle in a uv-coordinate sstem and a ra from the origin, which intersects the circle at point P. You can drag point P. A line is drawn tangent to the circle at P, intersecting the u-axis at point A and the v-axis at point B. A vertical segment from P intersects the u-axis at point C, and a horizontal segment from P intersects the v-axis at point D. Investigate 1. Use the properties of similar triangles to explain wh the following segment lengths are equal to the six function values of 5 maop: PA 5 tan PB 5 cot PC 5 sin PD 5 cos OA 5 sec OB 5 csc 2. The angle between the ra and the v-axis is the complement of, that is, it is Wh? Show that in each case the cofunction of θ is equal to the function of the complement of. 3. What happens to the six function values as changes? Describe how sine and cosine var as is made larger or smaller. Based on the figure, explain wh tangent and secant become infinite as approaches 90 + and wh cotangent and cosecant become infinite as approaches 0 +. dnamic precalculus exploration Precalculus with Trigonometr Course Sampler 151

20 Sketchpad Presentation Sketch Trigonometr Tracers This and other Sketchpad presentation sketches are available at to teachers who have purchased Precalculus with Trigonometr: Concepts and Applications. Shown here is the third page of the presentation sketch. The first two pages show a -value trigonometr tracer (which is a trace of the sine function as point C rotates) and an x-value trigonometr tracer (which is a trace of the cosine function as point C rotates). The third page, shown here, shows a tracer of x, the tangent function. presentation sketch You can find these teaching notes on the Notes page of the sketch: Press the buttons in order from top to bottom. Drag point C around at each stage to make appropriate observations. At the end, ou ma wish to animate C and then stop the animation to drag C manuall and make observations about the trigonometric function. Suggested questions: What are the maximum and minimum values of this function? How can ou explain these maximum and minimum values (and their corresponding angle measurements) in terms of the unit circle? For which values of is this function positive? For which values is it negative? Explain wh in terms of the unit circle. When is this function increasing? Decreasing? Explain wh in terms of the unit circle. 152 Precalculus with Trigonometr Course Sampler