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.)

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.)

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.)

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.

Angles In trigonometry, an angle is formed by rotating one ray (the initial side) to the other ray (the terminal side).

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.

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.

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.

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.

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.

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 6

Exercise 2 Which angle has a larger measure, ABC orDBE? neither is larger

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°.

We can totally have angles greater than 180°. 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!

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.

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.

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.

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

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

Radians

Radians for the Smart Masses

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?

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.

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.

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

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.

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.

Activity: Radians Use your ruler to draw a diameter from “9” to “3.” This is like the x-axis.

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.

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.

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.

Activity: Radians This should make sense since the circumference of a circle is 2πr, where r is the radius of the circle.

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.

2π Radians This time, we’re going to build a circle with various radian measures from 0 to 2π. Start by drawing a circle.

2π Radians Now 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.

2π Radians Next, 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.

2π Radians Finally, 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.

2π Radians So that concludes the radians you should have memorized. For the adventurous, you might try memorizing the twelths, too.

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 −315

Arc Measure and Arc Length The measure of an arc is the measure of the central angle it intercepts. It is measured in degrees.

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.

Arc Measure and Arc Length Arc measure = mC Amount of rotation Arc length: Actual length

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 −11π/6 −π/12 −2π 3 radians −4.5 radians