Contents 1 The Three Pictures of Trigonometry 1.1 The Three Pictures of Trigonometry . . . . . . . . . . . . . . 1.1.1 Angles and their measurement - Definitions . . . . . . 1.1.2 Conversions . . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 The definition of sine and cosine . . . . . . . . . . . . . 1.1.4 Tangent, Cotangent, Secant, and Cosecant . . . . . . . 1.2 The Graphs of Sine and Cosine . . . . . . . . . . . . . . . . . 1.2.1 Expansions, Contractions and Shifts . . . . . . . . . . 1.2.2 The slope of sin(x) at x = 0. . . . . . . . . . . . . . . . 1.2.3 The midline . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 The amplitude . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Frequency and period. . . . . . . . . . . . . . . . . . . 1.2.6 The period of sin(Cx) . . . . . . . . . . . . . . . . . . 1.2.7 Horizontal shifts . . . . . . . . . . . . . . . . . . . . . 1.3 Geometric Applications . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Triangle problems . . . . . . . . . . . . . . . . . . . . . 1.3.2 Triangle Definitions of Trigonometric Functions . . . . 1.3.3 The Law of Sines . . . . . . . . . . . . . . . . . . . . . 1.3.4 Why is the Law of Sines what it is? . . . . . . . . . . . 1.3.5 The Law of Cosines . . . . . . . . . . . . . . . . . . . . 1.3.6 Why is the Law of Cosines what it is? . . . . . . . . . 1.4 Trigonometric Equations . . . . . . . . . . . . . . . . . . . . . 1.5 Identities and the Return to Differentiation . . . . . . . . . . . 1.5.1 Polynomial Identities . . . . . . . . . . . . . . . . . . . 1.5.2 Symmetry Identities . . . . . . . . . . . . . . . . . . . 1.5.3 Circle Identities . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Addition Identities . . . . . . . . . . . . . . . . . . . . 1.5.5 The addition identity: why is it what it is? . . . . . . . 1.5.6 Identities Continued . . . . . . . . . . . . . . . . . . . 1.5.7 Double angle identities . . . . . . . . . . . . . . . . . . 1.5.8 Half angle identities . . . . . . . . . . . . . . . . . . . . 1.6 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 The Derivative of Sine . . . . . . . . . . . . . . . . . . 1.6.2 The Generalized Trigonometric Rules of Differentiation 1

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3 3 3 5 7 11 13 13 15 16 16 18 21 24 28 28 29 34 38 39 41 42 47 47 49 52 54 56 58 58 59 62 62 65

2

CONTENTS

1.7 1.8

1.6.3 Deriving the Derivative of Sine . . . . . . . . . . . . . . . . . Antidifferentiation and Integration . . . . . . . . . . . . . . . . . . . What the successful student will be able to do . . . . . . . . . . . . .

66 70 73

Chapter 1 The Three Pictures of Trigonometry Trigonometry for Calculus with Review

1.1

The Three Pictures of Trigonometry

There are Three Pictures of Trigonometry. Each picture captures a different aspect of trigonometry. The pictures are associated with the following topics. 1. Measuring angles 2. The definition of sine and cosine 3. The graphs of sine and cosine

1.1.1

Angles and their measurement - Definitions

Picture # 1 is associated with the measurement of angles. 3

4

CHAPTER 1. THE THREE PICTURES OF TRIGONOMETRY

Angles are measured as a fraction of the full circle. Figure 1.1: The measure of an angle l θ

x-axis

We have three examples. In the first and most familiar, the convention is that the full circle consists of 360o (degrees). If the blue arc represents 1/6-th of the full circle, then the measure of the blue arc in degrees is 1/6 th of 360o , or l = 360o ·

1 = 60o . 6

The measure of the full circle is up to the civilization that is measuring it. For example, I, leader of my new kingdom, choose to call the unit of angle measurement in my kingdom the zap. There are 51 zaps in a full circle. From then on, having made that choice, the size of an angle is measured in units of zaps. So the above angle is 1/6th of 51 zaps, or l = 51 zaps ·

1 = 8.5 zaps. 6

This is a rather unnatural unit of angle measurement, but would work perfectly well. A relatively natural unit of measurement assigns the full circle a size, or measure, of 2π, the circumference of a circle with unit radius. The unit for this measure of an angle is called the radian after the radius. In this case, the above blue angle (θ) would be 1/6th of 2π, or l = 2πradians ·

1 π = radians. 6 3

1.1. THE THREE PICTURES OF TRIGONOMETRY

1.1.2

5

Conversions

We have just measured l the length of the blue arc in three different units of measurement. If I have found l in one unit, there is an easy way to convert to the value of l in a different unit of measure. The conversion method is based on the fact that angles are measured as fractions of the full circle. So, arc length l (degrees) l (zaps) l (radians) = = = . full circle full circle in (degrees) full circle in (zaps) full circle in (radians)

Example 1 Suppose we have an angle with an arc that measures 67 degrees. 1. How many zaps would that angle measure? 2. How many radians would that angle measure? Solution to Part 1. We set up the following equality, where x is the unknown number of zaps. 67o x (zaps) = o 360 51 (zaps) Solving for x we get x (zaps) =

67o · 51 (zaps) 360o

x (zaps) =

51 (zaps) · 67o . 360o

or

Staring at this for a bit we realize that it will always work. If we start with an angle measuring y (degrees) and we want the convert that measurement to a measurement in the units of zaps, we do the following. x (zaps) =

51 (zaps) · y (degrees). 360o

6

CHAPTER 1. THE THREE PICTURES OF TRIGONOMETRY Solution to Part 2. We do exactly the same process, only the units and the

numbers change, the process remains the same. We set up the following equality, where x is the unknown number of radians. x (radians) 67o = 360o 2π (radians) Solving for x we get x (radians) =

67o · 2π (radians) 360o

x (radians) =

2π (radians) · 67o . 360o

or

Staring at this for a bit we realize that it will also always work. If we start with an angle measuring y (degrees) and we want the convert that measurement to a measurement in the units of radians, we do the following. x (radians) =

2π (radians) · y (degrees). 360o

Some of you have probably memorized this formula before. Me, I can’t remember it. I always forget it or get it upside-down. But I can remember the process, and thus don’t need to remember the formula. That said, when it comes times to take a quiz or an exam, investing a little time to memorize the formula will probably save a little time on the quiz or exam. But if you forget, well you can get what you need by equating the ratios, as we did here. Figure 1.2: The meaning of radian measure of an angle a

θ x-axis r

1.1. THE THREE PICTURES OF TRIGONOMETRY

7

Radians and the significance of the radius There is another aspect to the radian measure of angle. In Figure 1.2, if you measure the length of the blue arc; call that a, for arc length. And, if you measure the length of the radius; call that r, for radius. Then a θ (radians) = , r no matter what units of length you use to measure the arc length and the radius. In other words, measured in radians, the angle θ is measuring the length of the arc using the radius length to define the unit length.

1.1.3

The definition of sine and cosine

Picture # 2 is associated with the definition of sine.

Figure 1.3: The definition of sine and cosine sin(θ) θ cos(θ)

x-axis

When we want to answer the question, what is the sine of an angle θ, we use Figure 1.3. To describe what sin(θ) is, one finds the point where the ray corresponding the angle θ intersects the unit circle. On that intersection point one puts a big red dot. Then drawing a horizontal red line to the y-axis, we obtain a specific height, the y value where the dotted red line hits the y-axis. That height is sin(θ). Notice this is also the height of the red line segment in Figure 1.3. The cos(θ) is found by drawing a

8

CHAPTER 1. THE THREE PICTURES OF TRIGONOMETRY

green vertical line down to the x-axis. The value of x at which the green line intersects the x-axis is the cos(θ). The cos(θ) is also the length of the green line segment in Figure 1.3. That’s it. This doesn’t really tell us how to find the values of sine and cosine, it just tells us what the numbers mean. If you were really, really good with a pencil, you could draw a really careful circle and a really careful 60o arc, and a really careful red dot and a really careful horizontal line, and really carefully measure the height of the red line, and you would have the sin(60o ). Similarly for cos(60o ). Try it. In class we used a 45o triangle and an equilateral triangle to work out the values of sin(θ) for θ equal to 30o , 60o , and 90o . In addition, we used Figure 1.3 to work out sin(θ) for θ equal to 0 and 90o . We then used symmetry to extend this information to those values of θ between 90o and 180o . When we were done, we had the following table of data.

θo

0 30 45 60 90 120 135 150 180

θ (rad) 0 sin(θ)

0

π 6

π 4

1 2

√1 2

π 3 √ 3 2

π 2

2π 3 √ 3 2

1

3π 4

5π 6

π

√1 2

1 2

0

(1.1)

By using symmetry we can extend this to the angles between 180o and 360o to get the following table of data.

θo

180 210

θ (rad)

π

7π 6

sin(θ)

0

− 21

225

240

270

300

315

5π 4

4π 3 √

3π 2

5π 3 √

7π 4

11π 6



− √12

− 12

0

− √12 −

3 2

−1 −

3 2

330 360 (1.2)

Finally, using the same triangles and the same ideas of symmetry we can arrive

1.1. THE THREE PICTURES OF TRIGONOMETRY

9

at a similar table of data for the cosine function.

θo

0 30 45 60 90 120

θ (rad)

0

cos(θ)

1

π 6 √ 3 2

π 4

π 3

π 2

2π 3

√1 2

1 2

0

− 12

135

150

180

210

225

3π 4

5π 6 √

π

7π 6 √ − 23

5π 4

4π 3

3π 2

5π 3

7π 4

− √12

− 12

0

1 2

√1 2

− √12 −

3 2

−1

240 270 300 315 330 360

(1.3)

OH MY GOD! No, really it’s not that bad. You don’t have to memorize it. Just know how to use the triangles to construct the value2, as we did in class.

11π 6 √ 3 2

2π 1

10

CHAPTER 1. THE THREE PICTURES OF TRIGONOMETRY

Problems. 1. For the angle θ depicted in the figure to the right, what is the exact value of sin(θ)? (-0.6,0.8) θ

2. Convert 137o to radians. 3. Convert 247o to radians (Round to 2 decimal places). 4. A DVD rotates 5/6 of a revolution. How many radians has it travelled? 5. In a circle with radius of 6 yards find the radian measure of the central angle whose terminal side intersects the circle forming an arc which measures 8 yards in length. 8

6

6. What are the exact values of the following. a. cos(30o ) = b. sin(45o ) = c. cos(60o ) = d. sin(30o ) = e. cos(270o ) = o

f. sin(90 ) = o

g. cos(150 ) =

o

h. sin(135 ) =

n. sin( 5π )= 4

i. cos(225o ) =

o. cos( 3π )= 4

j. sin(330o ) =

p. sin( 5π )= 6

k. cos(270o ) =

q. cos( 11π )= 6

l. sin(150o ) =

r. sin(2π) =

m. cos( π3 ) =

s. cos(π) =

1.1. THE THREE PICTURES OF TRIGONOMETRY

1.1.4

11

Tangent, Cotangent, Secant, and Cosecant

We also define four more functions directly from the graph of the unit circle just like we did for sine and cosine. See Figure 1.4. Figure 1.4: The definition of tangent, cotangent, secant and cosecant y θ x

tan(θ) =

y x

cot(θ) =

x y

x-axis

sec(θ) =

1 x

csc(θ) =

1 y

These functions can be related to the values of sine and cosine as follows. sin(θ) cos(θ) 1 sec(θ) = cos(θ)

tan(θ) =

cos(θ) sin(θ) 1 csc(θ) = sin(θ)

cot(θ) =

These relationships are typically used to determine the values of these functions and to understand their properties. Example 2 Find the exact value of tan(π/4). Answer: Because tan(θ) =

sin(θ) we have cos(θ)

sin(π/4) = tan(π/4) = cos(π/4)

√1 2 1 √ 2

= 1.

12

CHAPTER 1. THE THREE PICTURES OF TRIGONOMETRY Problems.

What are the exact values of the following.

1. tan(30o ) = 2π )= 3

2. sec(30o ) =

7. tan(

3. sec(135o ) =

8. cot(

3π )= 4

9. sec(

2π )= 3

4. tan(120o ) 5. tan(240o ) = o

6. csc(240 ) =

10. cot(

4π )= 3

1.2. THE GRAPHS OF SINE AND COSINE

13

Trigonometry for Calculus with Review -

Part 2

-

1.2

The Graphs of Sine and Cosine

1.2.1

Expansions, Contractions and Shifts

Our goal is now to explore the third picture of trigonometry, the graph of sin(x). In addition to examining the graph of sin(x), we will explore how we can get new graphs from the graph of sin(x) by expansions, contractions, and shifts. One motivation is the pendulum. Figure 1.5: Measuring the position of a pendulum

x

x-axis

x

x

In class we may have measured the position of a pendulum with our tricorder. It would have displayed data like that in Figure 1.6.

14

CHAPTER 1. THE THREE PICTURES OF TRIGONOMETRY Figure 1.6: Tricorder Pendulum Data 3

Pendulum Data

Position (m)

2.5

2

1.5

1

0.5

0

0

2

4

6

8

Time (sec)

10

12

We would like to model this data with a sin(x) function using some combination of expansion, contraction, and or shifting. To begin we examine the graph of the function sin(x). Earlier we constructed a table of data on the sine function. Figure 1.7 depicts a graph of that data.

Figure 1.7: Sine Data 2 sin(x)

1.5 1

− π2 − 3π 2 −π

0.5 0

3π 2

π π 2

−0.5 −1 −1.5 −2

−6

−4

−2

0

2

4

6

If we struggle to get the values for more points (and how might we do that?), we can extend the graph to get a shape that seems to look more and more like that in Figure 1.8. When we address the derivative of sin(x) and cos(x) we will see further confirmation that the shape in Figure 1.8 is the correct graph.

1.2. THE GRAPHS OF SINE AND COSINE

15

Figure 1.8: The Graph of sin(x) 2 sin(x)

1.5 1

− π2 − 3π 2 −π

0.5 0

3π 2

π π 2

−0.5 −1 −1.5 −2

1.2.2

−6

−4

−2

0

2

4

6

The slope of sin(x) at x = 0.

In class we may have discussed further evidence that Figure 1.8 is the correct shape for the graph by exploring what we expected the slope of the graph to be at x = 0, that is, when the angle is zero. We would have done this by examining the diagram used to define sin(x). In the right hand side of Figure 1.9 we see that as the angle x sin(x) x→0 x

Figure 1.9: Considering lim

sin(θ) θ cos(θ)

x-axis

sin(θ)

θ

x-axis

get small, the ratio of sin(x) (the length of the red, vertical line segment) to x (angle measured in radians, the blue arclength) is getting closer and closer to 1 - because the two lengths are getting closer and closer. (It is true, both are getting smaller and

16

CHAPTER 1. THE THREE PICTURES OF TRIGONOMETRY

smaller, but their lengths are getting closer even faster.) This confirms what we see in Figure 1.8, where the slope of the graph of sin(x) at zero is positive. If we were to rescale the figure, so that 1 unit in the x direction and 1 unit in the y direction were the same size, we would see the slope would actually look like it equaled 1.

1.2.3

The midline

When we discussed the midline, the center of sin(x) we saw that the function f (x) = A + sin(x)

(1.4)

has its midline lifted A units, which means down if A is negative. See Figure 1.10. We say the graph of f (x) = A + sin(x) is a vertical shift of the graph of sin(x). Figure 1.10: sin(x) versus A + sin(x) 4

A+sin(x) 3

sin(x)

A 2

A

1

0

−1

−2

1.2.4

−6

−4

−2

0

2

4

6

The amplitude

In contrast, there is amplitude which measures the distance between the midline and the extremes, the maximum and minimum. Compare the graphs of sin(x) and B sin(x) as seen in Figure 1.11. As we discussed in class the amplitude of sin(x)

1.2. THE GRAPHS OF SINE AND COSINE

17

is 1, and the amplitude of B sin(x) is |B| (the absolute value, because B could be negative). If |B| > 1, then B sin(x) is called a vertical expansion of sin(x). If |B| < 1, then B sin(x) is called a vertical contraction of sin(x). Figure 1.11: sin(x) versus B sin(x) 3

3

2

A

B sin(x)

For B>0

B

For B