Name: Group Members: Exploration 6-1a: Sine and Cosine...

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Precalculus with Trigonometry Course Sampler 133

170 Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book© 2012 Key Curriculum Press

Name: Group Members:

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

Objective: Find the shape of sine and cosine graphs by plotting them on graph paper.

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90° 180° 270° 360° 450° 540° 630° 720°

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90° 180° 270° 360° 450° 540° 630° 720°

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134 Precalculus with Trigonometry Course Sampler

172 Exploration Masters Precalculus with Trigonometry: Instructor’s Resource Book


Exploration 6-2a: Transformed Sinusoid Graphs Date:

Objective: Given the equation for a transformed sinusoid, sketch the graph, and vice versa.

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© 2012 Key Curriculum Press

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Exploration 6-3a: Tangent and Secant Graphs Date:

Objective: Discover what the tangent and secant function graphs look like and how they relate to sine and cosine.

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Precalculus with Trigonometry: Instructor’s Resource Book Exploration Masters 175


Exploration 6-3b: Transformed Tangent Date: and Secant GraphsObjective: Sketch transformed tangent, cotangent, secant, and cosecant graphs, and find equations from given graphs.

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Exploration 6-7a: Oil Well Problem Date:

Objective: Use sinusoids to predict events in the real world.

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100 65 30

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Fence

y = 2000 ft

x = 700 ft

xInaccessible land Available land


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Precalculus with Trigonometry: Instructor’s Resource Book Exploration Masters 185


Exploration 6-8b: Motorcycle Problem Date:

Objective: Find angular and linear velocities of connected rotating objects.


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348 CAS Activities Precalculus with Trigonometry: Instructor’s Resource Book© 2012 Key Curriculum Press


CAS Activity 6-4a: Inverse Trigonometric Functions Date:

Objective: Prove co t –1 x = ta n –1 1 __ x and its parallel secant and cosecant forms. Explain why only three inverse trigonometric functions are required.

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Precalculus with Trigonometry: Instructor’s Resource Book CAS Activities 349© 2012 Key Curriculum Press

Define dist(a,b,c,d) = _____________

Zeros

Zeros(dist(0, 0, x, y) dist(432, 7, x, y) 98.4 0, y)

Zeros


CAS Activity 6-7a: Epicenter of an Earthquake Date:

Objective: Discover the minimum number of points required to definitively locate the source of an earthquake.

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Precalculus with Trigonometry: Instructor’s Resource Book The Geometer's Sketchpad® Activities 417© 2012 Key Curriculum Press

© 2009 Key Curriculum Press 1

Transformations of Circular Functions

In this activity you will use a

point on the unit circle to

construct dilated images of

circular functions.

SKETCH AND INVESTIGATE

1. Open Circular Transforms.gsp. The sketch contains a parameter k that

currently equals 2. Use the Calculator to multiply k by the angle measure of DC.

2. Mark point A as the center of rotation using Transform Mark Center. Similarly, mark the calculation from step 1 as the angle of rotation.

3. Rotate point D by selecting it and choosing Transform Rotate. 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 every complete trip that point C makes around the circle, how many 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 you built in the activity

Trigonometry Tracers, traces out sin(mDC ). Press the Animate Point C button

to watch point E in action.

Q4 You’re about to create the graph of sin(k mDC . Before you do, make a

prediction: Based on your answers to Q2 and Q3, what do you predict the

graph will look like?

5. Measure yF by selecting point F and choosing Measure Ordinates (y).

6. Plot the point mDC, yF

by selecting in order mDC and yF, and then choosing

Graph Plot as (x, y).

7. Label the plotted point G, and turn on tracing for it.

Q5 Animate C, and observe the trace of G. Is your prediction about the graph of

sin(k mDC correct?

8. Change the value of parameter k to draw new sine curves.

Q6 By taking new measurements, create the graphs of cos(k mDC and

tan(k mDC . Describe the appearance of each of these functions.

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DA

C

To mark the calculation as the angle of rotation, select it and choose Transform Mark Angle.

Choose Display Erase Traces to erase existing traces.

To turn on tracing, select the point and choose Display Trace Point.

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418 The Geometer's Sketchpad® Activities Precalculus with Trigonometry: Instructor’s Resource Book© 2012 Key Curriculum Press

© 2009 Key Curriculum Press 1

SKETCH AND INVESTIGATE

Q1 DAF is twice as large as DAC.

Q2 Point F travels twice around the circle for every 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 every revolution of C.

Q4 Predictions will vary. 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 cycles

between 0 and 2 .

8. When you change k, the period becomes 2 /k, and the graph shows k

complete cycles between 0 and 2 .

Q6 The graphs of these functions resemble the graphs produced in the

Trigonometry Tracers activity, but (like the sine plot) compressed in the

x direction so that they show k cycles between 0 and 2 .

EXTENSION

You could challenge students to fi nd a way to modify 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 y-coordinate as a function of

the angle. If segment AF is constructed as a ray, it’s possible to produce both

compression and stretching. Alternatively, you could dilate point F toward

or away from center point A.

PRESENT

To present this activity to the whole class, use Circular Transforms Present.gsp

Transformations of Circular Functions ACTIVITY NOTES

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Precalculus with Trigonometry: Instructor’s Resource Book Blackline Masters 31© 2012 Key Curriculum Press


Problem Set 6-2/Pages 292–293 Date:

20°25°70° 65° 110° 155° 200°

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Precalculus with Trigonometry: Instructor’s Resource Book Blackline Masters 33© 2012 Key Curriculum Press


Problem Set 6-4/Pages 305–306 Date:

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Precalculus with Trigonometry: Assessment Resources Section, Chapter, and Cumulative Tests 61© 2012 Key Curriculum Press

Part 1: No calculators allowed (1–9)

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Name: Date:

Test 15, Sections 6-1 to 6-3 Form AObjective: Draw graphs of sinusoids and of tangent and secant functions.

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Part 2: Graphing calculators allowed (10–24)

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62 Section, Chapter, and Cumulative Tests Precalculus with Trigonometry: Assessment Resources© 2012 Key Curriculum Press

Name: Date:

Test 15, Sections 6-1 to 6-3 continued Form A

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Precalculus with Trigonometry: Assessment Resources Section, Chapter, and Cumulative Tests 63© 2012 Key Curriculum Press

Part 1: No calculators allowed (1–9)

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Name: Date:

Test 15, Sections 6-1 to 6-3 Form BObjective: Draw graphs of sinusoids and of tangent and secant functions.

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Part 2: Graphing calculators allowed (10–24)

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64 Section, Chapter, and Cumulative Tests Precalculus with Trigonometry: Assessment Resources© 2012 Key Curriculum Press

Name: Date:

Test 15, Sections 6-1 to 6-3 continued Form B

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Problem Set 6-1

2

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Problem Set 6-2

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Precalculus with Trigonometry: Solutions Manual Problem Set 6-2 87© 2012 Key Curriculum Press

Chapter 6 Applications of Trigonometric and Circular Functions

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88 Problem Set 6-2 Precalculus with Trigonometry: Solutions Manual© 2012 Key Curriculum Press

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Dynamic Precalculus Exploration

Experience the online version of this exploration at www.keymath.com/precalc.

Variation of Tangent and Secant

The sketch below will help you understand how the functions tangent and secant vary as their arguments vary.

Sketch

This sketch shows a unit circle in a uv-coordinate system and a ray 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 why 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 ray and the v-axis is the complement of , that is, it is 90° 2 . Why? 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 vary as is made larger or smaller. Based on the figure, explain why tangent and secant become infinite as approaches 90+ and why cotangent and cosecant become infinite as approaches 0+.

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Sketchpad Presentation Sketch

Trigonometry Tracers

This and other Sketchpad presentation sketches are available at www.keymath.com/keyonline to teachers who have purchased Precalculus with Trigonometry: Concepts and Applications.

Shown here is the third page of the presentation sketch. The first two pages show a y-value trigonometry tracer (which is a trace of the sine function as point C rotates) and an x-value trigonometry tracer (which is a trace of the cosine function as point C rotates). The third page, shown here, shows a tracer of y __ x, the tangent function.

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, you may wish to animate C and then stop the animation to drag C manually and make observations about the trigonometric function. Suggested questions:

• Whatarethemaximumandminimumvaluesofthisfunction?

• Howcanyouexplainthesemaximumandminimumvalues(andtheircorrespondingangle measurements) in terms of the unit circle?

• Forwhichvaluesof is this function positive? For which values is it negative? Explain why in terms of the unit circle.

• Whenisthisfunctionincreasing?Decreasing?Explainwhyintermsoftheunitcircle.

Name: Group Members: Exploration 6-1a: Sine and Cosine...

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