Precalculus 115, section 5.1 The Unit Circle & Radian Measure of Angles notes by Tim Pilachowski

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1 Precalculus 5, section 5. The Unit Circle & Radian Measure of Angles notes b Tim Pilachowski We ll be working with the unit circle, with radius equal to. On a Cartesian grid we ll place the center of the circle at the origin. Note first the smmetries of the unit circle: horizontal, vertical and across the origin. If we know coordinates of a point (a, b) in QI, then we know the coordinates of smmetric points in the other three quadrants. Note that an point P on the unit circle with coordinates (, ) can be used to form a right triangle, b starting at the origin and moving over and up/down (forming the right angle). The hpotenuse is a radius of the circle. P(, ) P(, ) P(, ) Using the Pthagorean formula, we can derive the equation of the unit circle: + =. We ll be using this basic identit a lot in chapters 5 through 7. Eample A. Find P(,) given > 0 and =. Eample B. Find P(,) given = and P in QIII. We now move to a consideration of angles, a geometric and trigonometric figure consisting of a verte and two sides: the initial side and the terminal side. We ll place the verte at the center of the unit circle. B convention, the initial side will be drawn horizontall to the right along the -ais. The terminal side will connect the origin with a terminal point P on the unit circle. verte terminal side initial side

2 In elementar and high school Geometr, ou would likel have measured angles with degrees, minutes and seconds a method devised in ancient Bablon. Angle measures such as 0º, 0º, 5º, 0º, 90º (right angle), 80º (straight angle), and 0º (full circle) would have been often-used. For purposes of Trigonometr and Calculus, we ll find it much more convenient to measure angles in terms of radians. To define radians, we put our angle into a circle so that the verte is located at the center, and designate the radius of the circle equal to. On a Cartesian grid we ll place the (verte of the angle) = (center of the circle) at the origin. This unit circle has a circumference (formula C = r) equal to. integer Some quick notes on the number. It is an irrational number, i.e. it cannot be written as an fraction, integer and as a decimal it is non-terminating and non-repeating. (See Pilachowski s Rules of Mathematics. and.59.) It is an eact value when written as. and 7 are approimations. In this class it will be standard procedure to give the eact value as our answer. Back to the unit circle. Its circumference is equal to. Radian measure is defined as the distance around the circumference one must travel to get from the initial side of the angle to the terminal side. If one travels all the wa around the unit circle, it would be an angle of 0º and radians. This relationship gives us the conversion factor to convert degrees to radians and back again: 0 degrees = radians and radians = degrees. 0 Note that the tet uses the simplest form fractions 80 and. 80 Eample C: Convert 0º, 0º, 5º, 0º, 90º and 80º to radians. answers: 0,,,,, ; Note that and radians, when placed on the unit circle, match what we know about right and straight angles. Eample D: Convert 7, and radians to degrees. answers: 5º, 0º, 70º

3 Note that in the latter two answers to Eample D, we came up with angles not seen in Geometr, i.e angles larger than 80º. (This is just one one of the reasons wh radians will be more useful and convenient than degrees.) Eample E: Find P(,) for angle. The convention is that we will also designate the direction of travel as follows: couterclockwise is positive movement, and clockwise is negative movement. 5 Note: A central angle of radians puts us at the same place as + radians. We ll call these two angles coterminal. We can also travel more than once around the circle, with each transit taking us radians. So 8 = + = + is coterminal with, 9 as well as with = + = +, and so on. 5 8 Likewise, = = is also coterminal with, however it is ver important to note that its direction of travel is clockwise rather than counterclockwise. Eample E: Find P(,) for angle 9.

4 P(,) for angle P(,) for angle P(,) for angle Reference angles. You ll need to know the following values for angles in Quadrant I. t 0 If we know P(,) for a reference angle t in QI, then we can find P(,) for an angle in the other three quadrants.

5 t = The angle is also the reference angle for all of the angles smmetric to it in QIII and QIV. The angle is also the reference angle for all of the angles smmetric to it in QIII and QIV. 5 and The angle is also the reference angle for all of the angles smmetric to it in QIII and QIV.