lecture02 - 1 1.1 Trigonometry Angles I mentioned on...

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1 Trigonometry 1.1 Angles I mentioned on Tuesday that we don’t use degrees in calculus. Instead we use a unit called radians. Definition 1. One radian is the angle that gives an arc length equal to the radius. Since the circumference of a circle is 2 πr , there are 2 π radians in one revo- lution of the circle. However, there are also 360 in one revolution. This gives us a very nice formula to go back and forth: 2 π rad = 360 Divide both sides by 2 to get the formula π rad = 180 Example 1.1. Find the radian measure of 60 and find the degree measure of 5 π 4 OK, so now we have a new way of indicating angles. So what? Why go through all this work? Well, using radians makes some calculations easier. For example, look at the length of an arc of a circle with angle θ Picture goes here Using some geometry, we can get the formula θ 2 π = a 2 πr and so the length of the arc is expressed as a = . Example 1.2. If the radius of a circle is 5cm, what angle gives an arc of length 6cm? Example 1.3. If a circle has radius 3cm, what is the length of the arc given by the angle 3 π 8 ? Now, to draw an angle, we use something called the standard position. This means that we start on the positive x axis and rotate counterclockwise if θ > 0 and clockwise if θ < 0. More pictures. Woohoo Note: Some angles may look like they have the exact same standard position.
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