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The Greeks doing trig in fancy dress Intro to Trigonometry For those later influenced by Greek mathematics trigonometry deals with periodic functions whose graphs are usually pretty wavy and whose applications far exceed the polygonal constraints of triangles periodic motion sound and ot ID: 719001

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Slide1

Intro to Trigonometry

For the Greeks, trigonometry dealt with the side and angle measures of triangles, practical math for building stuff and locating the position of a particular constellation for accurately navigating their boats and dubiously predicting their future.

(The Greeks doing trig in fancy dress.)Slide2

Intro to Trigonometry

For those later influenced by Greek mathematics, trigonometry deals with periodic functions, whose graphs are usually pretty wavy, and whose applications far exceed the polygonal constraints of triangles: periodic motion, sound, and other interesting stuff.

(An expensive piece of equipment that your iPhone could probably duplicate for 99 cents.)Slide3

Intro to Trigonometry

For those later influenced by Greek mathematics, trigonometry deals with periodic functions, whose graphs are usually pretty wavy, and whose applications far exceed the polygonal constraints of triangles: periodic motion, sound, and other interesting stuff.

(Trig is instrumental in solving crimes.)Slide4

Angles

Recall from geometry that an angle

is made from two rays joined at a common endpoint called a vertex.We usually use the Greek letters

α

(alpha),

β

(beta), and

θ

(theta) to label an angle. Or maybe just a capital letter.Slide5

Angles

In trigonometry, an angle is formed by rotating one ray (the initial side

) to the other ray (the terminal side).Slide6

Standard Position

It is often convenient and fun to place angles in a rectangular coordinate system.

An angle is in standard position if its vertex is on the origin and its initial side is on the positive x-axis.Slide7

Positive vs. Negative

An angle is positive

if the terminal side is rotated counter-clockwise.An angle is negative if the terminal side is rotated clockwise.Slide8

The Measure of an Angle

The measure of an angle is the amount of rotation from the initial side to the terminal side as measured in

degrees or radians.Slide9

Five Days Short…

The curious among you may be wondering why there are 360 degrees in a circle, while others may not even care. The answer is actually pretty simple: It’s because there are 360 days in the year. At least that’s what the Babylonians thought, and they are the ones who came up with the crazy idea called a degree.Slide10

Five Days Short…

Each year, of course, is made up of 12 “months.” Further, each of those “months” is made up of 30 “days.” 12 times 30 equals 360 degrees, I mean days.Slide11

Exercise 1

On a clock, how many degrees does the hour hand rotate each hour?How many degrees does the minute hand rotate each minute?

30 degrees

6Slide12

Exercise 2

Which angle has a larger measure, ABC orDBE?

neither is largerSlide13

More on Degrees

In geometry, we always measured angles by the smallest amount of rotation from the initial side to the terminal side, from 0° to 180°. Slide14

More on Degrees

That’s because we were mostly dealing with triangles and stuff. But we can just as easily have angles with measures greater than 180°, just ask Danny Way.

We can totally have angles greater than 180°.

Now check this out!Slide15

360 Degrees

So now, let’s build ourselves a circle with various degree measures from 0 to 360. Yes, before you ask, you will have to have these memorized.

Start by drawing a circle.Slide16

360 Degrees

Now break the circle into fourths by adding an x- and

y-axis. Label the points of intersection 0, 90, 180, 270 and 360.Slide17

360 Degrees

Next, add your clock hours. We already have 3, 12, 9, and 6. Just put two equally spaced marks between each quarter. Label these marks with multiples of 30.Slide18

360 Degrees

Finally, add a mark in the middle between 30 and 60. This is 45. Now add your multiples of 45.Slide19

360 Degrees

So that concludes the degrees you should have memorized.For the adventurous, you might try memorizing multiples of 15, too.Slide20

RadiansSlide21

Slide22

Radians for the Smart MassesSlide23

Activity: Radians

Here’s an interesting question: If you were to take the radius of a circle and wrap it around the circle’s circumference, how far would it reach?Slide24

Activity: Radians

Use a ruler to draw a radius from the center of the circle to the “3.” This is like the initial side of an angle in standard position.Slide25

Activity: Radians

Now cut a thin strip of paper from the bottom edge of your paper and mark the length of the radius of your circle on the left side of the strip.Slide26

Activity: Radians

Carefully wrap this length along the circumference of the circle and mark it with your pencil.Slide27

Activity: Radians

Use your ruler to connect this mark to the center of the circle with another radius. This is the terminal side of an angle we’ll call θ.Slide28

Activity: Radians

The arc that intercepts θ has length 1

radian, so we say the measure of θ = 1 radian.Approximately how many degrees is 1 radian?

Now let’s see how many radians it takes to span the circle.Slide29

Activity: Radians

Use your ruler to draw a diameter from “9” to “3.” This is like the x

-axis.Slide30

Activity: Radians

Now use a compass to measure the radian arc length. Copy this length around your circle multiple times until you have gone (nearly) all the way around.Slide31

Activity: Radians

You should notice that it takes a little bit more than 3 radians to span a semicircle. In fact, it takes exactly π (≈3.14) radians.Slide32

Activity: Radians

Also notice that it takes a bit more than 6 radians to span the full circle, which is exactly 2π (≈

6.28) radians.Slide33

Activity: Radians

This should make sense since the circumference of a circle is 2π

r, where r is the radius of the circle. Slide34

Radian: Definition

One radian

is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle.Slide35

2π

RadiansThis time, we’re going to build a circle with various radian measures from 0 to 2π.

Start by drawing a circle.Slide36

2π

RadiansNow break your circle into fourths by adding an

x- and y-axis. Since a semicircle is π, the quarter circle is π/2. Now count by halves.Slide37

2π

RadiansNext, add your clock hours. We already have 3, 12, 9, and 6. Just put two equally spaced marks between each quarter. Since there are 6 of these marks on the semicircle, count by sixths.Slide38

2π

RadiansFinally, add a mark in the middle between π/6 and π/3. Since there would be four of these along the semicircle, these must be fourths.Slide39

2π

RadiansSo that concludes the radians you should have memorized.

For the adventurous, you might try memorizing the twelths, too.Slide40

Exercise 3

Sketch each of the following angle measurements. To do so, start by drawing a circle with the x- and

y-axes, and then put your angle into standard position.

60

135

210

270

−15

−150

−275

−315Slide41

Arc Measure and Arc Length

The measure of an arc is the measure of the central angle it intercepts. It is measured in degrees.Slide42

Arc Measure and Arc Length

An arc length is a portion of the circumference of a circle. It is measured in linear units and can be found using the measure of the arc.Slide43

Arc Measure and Arc Length

Arc measure = mCAmount of rotation

Arc length:Actual lengthSlide44

Exercise 4

Sketch each of the following angle measurements. To do so, start by drawing a circle with the x- and

y-axes, and then put your angle into standard position.

π/3

3π/4

−

π/2

−11

π

/6

−

π/12

−2

π

3 radians

−4.5 radians