MAT 170 Pre-Calculus Notes on Radian Measure Radian Angles Terri L. Miller Spring 009 revised April 17, 009 1. Radian Measure Recall that a unit circle is the circle centered at the origin with a radius of one. Since the radius is one, the distance around the circle, the circumference, is π. The terminal side of an angle cuts the unit circle at a point, P (x, y). The measure of the distance in moving from (1, 0) to this point, P (x, y) along the unit circle in a counterclockwise direction is the radian measure of the angle, a positive number. Note: any real number can be a radian measure! So a negative number would indicate to move the distance from (1, 0) to the point in a clockwise direction. For the unit circles pictured below, we have indicated by a dot, the point P (x, y) where the terminal side of an angle would cut the circle. The angle is shaded and the measurement indicated. Figure 1. The angles of radian measures 1 and Here C marks the terminal point on the unit circle for the angle α = 1 and D marks the terminal point on the unit circle for the angle β =. Here D marks the terminal point on the unit circle for the angle θ = 4 and E marks the terminal point on the unit circle for the angle δ = π 5. Let us consider some standard angles on the unit circle. First, recall that the distance around the unit circle is π. Hence θ = π is one full circuit of the unit circle so that both the initial and terminal points are (1, 0). Going half way around the circle would be a distance of π = π, so the terminal point would be ( 1, 0). Here C marks the terminal point on the unit circle for the angle θ = π and B marks the terminal point on the unit circle for the angle α = π.
Figure. The angles of radian measures 4 and π 5 Figure 3. The angle π and the angle π The standard angles to mark off are the multiples of 0, π 6, π 4, π 3, π, and π. These are chosen since we do know the exact coordinates of their terminal points. However, we will save the coordinates for a bit later. Figure 4. Some standard angles. c 009 ASU School of Mathematical & Statistical Sciences and Terri L. Miller
The angles pictured here are α = π 6, β = π 4, γ = π 3, δ = π 3, ɛ = 7π 6, and δ = 7π 4. Here are some more standard but negative angles. Figure 5. Some negative angles. The angles pictured here are α = π 4, β = π, γ = 5π 6, δ = 5π 4, and ɛ = 5π 3. We can also mark off any real number. Figure 6. Arbitrary numbers The angles pictured here are α = 0.5, β =.7, γ = 15, and δ = 3. 1 3 c 009 ASU School of Mathematical & Statistical Sciences and Terri L. Miller
. The Coordinates We know the coordinates for all of the points along the axes. Further, we know that every point on the unit circle satisfies the equation x + y = 1. The angle π cuts the unit circle in 4 the middle of the first quadrant. This corresponds to the line y = x, The coordinates of this point can easily be obtained by solving the equations y = x and x + y = 1 for x. When ( ) this is done, we find the coordinates are,. Figure 7. Coordinates on the Unit Circle In the left hand figure, the points along the axes are given. In the right hand figure, the points of intersection with the lines y = x and y = x have been marked. The coordinates ( ) ( ) ( ) for these point are F =,, G =,, H =,, and J = ( ),. The use of equilateral triangles and some basic geometry yields the coordinates for the points corresponding to the terminal sides of the angles π and π and hence their multiples in other 6 3 quadrants. These are all labelled later on a single unit circle. 3. The Functions The cosine is the x coordinate of the point on the unit circle corresponding to the terminal side of an angle. The function is written cos θ. The sine is the y coordinate of the point on the unit circle corresponding to the terminal side of an angle. The sine function is written sin θ. The other four trigonometric functions are defined in terms of the sine and cosine, the tangent is the ration of the sine over the cosine (denoted tan θ), the secant is the reciprocal of the cosine (denoted sec θ), the cosecant is the reciprocal of the sine (denoted csc θ), and the cotangent is the ratio of the cosine over the sine (denoted cot θ). 4 c 009 ASU School of Mathematical & Statistical Sciences and Terri L. Miller
Figure 8. The Unit Circle 4. Links For further reading, I recommend checking out the following url: 5 c 009 ASU School of Mathematical & Statistical Sciences and Terri L. Miller
http://www.themathpage.com/atrig/radian-measure.htm For further unit circle help, try the following url: http://members.shaw.ca/ron.blond/tle/wrapping.function.applet/index.html or http://www.mathlearning.net/learningtools/flash/unitcircle/unitcircle.html 6 c 009 ASU School of Mathematical & Statistical Sciences and Terri L. Miller