Chapter 6: Radians

Introduction to Trig Radians

Introduction

A tablet of ancient symbols created by the Babylonians
Figure 1
The Babylonians used a base-60 number system because the number 60 has many factors. They did not invent decimal fractions, so they found it difficult to deal with remainders when doing division. But 60 can be divided evenly by 2, 3, 4, 5, and 6, which made calculations with common fractions much easier.
In geometry, Babylonian mathematicians used the corner of an equilateral triangle as their basic unit of angular measure, and naturally divided that angle into 60 smaller angles (Figure 2). Now, if the corners of six equilateral triangles are placed together, they form a complete circle, and that is why there are six times 60, or 360 degrees of arc in a circle.
The corners of six equilateral triangles placed together showing they form a complete circle
Figure 2
But why do we need another, different way to measure angles? In this chapter we’ll study radian measure, which at first may seem awkward and unnatural (Figure 3). As a hint, consider that although 360 is not fundamental to circles, the number [latex]\pi[/latex] is.
Different polygons showing angles with radian measures
Figure 3
Radians connect the measure of an angle with the arclength it cuts out on a circle.
A circle of radius 1unit rolling along a straight line showing a distance of pi and 2pi.
Figure 4
In calculus and most other branches of mathematics beyond practical geometry, angles are nearly always measured in radians. Because radians arise naturally when dealing with circles, important relationships are expressed more concisely in radians. In particular, results involving trigonometric functions are simpler when the variables are expressed in radians.

Class Activities

Activity 6.2. Radians

Here is a unit circle with arclengths labeled, measured counterclockwise from [latex](1,0){.}[/latex] (Note that the distance around the whole circle is 6.28 units!)

grid

  1. Use the unit circle to estimate the sine, cosine, and tangent of each angle in radians.
    1. [latex]0.6[/latex]
    2. [latex]2.3[/latex]
    3. [latex]3.5[/latex]
    4. [latex]5.3[/latex]
  2. Use the unit circle to estimate two solutions to each equation.
    1. [latex]\cos t = 0.3[/latex]
    2. [latex]\sin t = 0.7[/latex]
  3. Sketch the angle on the unit circle. Find the reference angle in radians, rounded to two decimal places, and sketch the reference triangle.
    1. [latex]1.8[/latex]
    2. [latex]5.2[/latex]
    3. [latex]3.7[/latex]
  4. Give a decimal approximation to two places for each angle, then the degree measure of each.
    Radians [latex]0[/latex] [latex]\dfrac{\pi}{12}[/latex] [latex]\dfrac{\pi}{6}[/latex] [latex]\dfrac{\pi}{4}[/latex] [latex]\dfrac{\pi}{3}[/latex] [latex]\dfrac{5\pi}{12}[/latex] [latex]\dfrac{\pi}{2}[/latex] [latex]\dfrac{7\pi}{12}[/latex] [latex]\dfrac{2\pi}{3}[/latex] [latex]\dfrac{3\pi}{4}[/latex] [latex]\dfrac{5\pi}{6}[/latex] [latex]\dfrac{11\pi}{12}[/latex] [latex]\pi[/latex]
    Decimal
    Approx.
    [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{0}[/latex]
    Degrees [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{000}[/latex] [latex]\hphantom{0}[/latex]
    Radians [latex]\pi[/latex] [latex]\frac{13\pi}{12}[/latex] [latex]\dfrac{7\pi}{6}[/latex] [latex]\dfrac{5\pi}{4}[/latex] [latex]\dfrac{4\pi}{3}[/latex] [latex]\dfrac{17\pi}{12}[/latex] [latex]\dfrac{3\pi}{2}[/latex] [latex]\frac{19\pi}{12}[/latex] [latex]\dfrac{5\pi}{3}[/latex] [latex]\dfrac{7\pi}{4}[/latex] [latex]\frac{11\pi}{6}[/latex] [latex]\frac{23\pi}{12}[/latex] [latex]2\pi[/latex]
    Decimal
    Approx.
    [latex]\hphantom{0}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{0}[/latex]
    Degrees [latex]\hphantom{0}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{00}[/latex] [latex]\hphantom{0}[/latex]
  5. On the unit circle above, plot the endpoint of each arc in standard position.
    1. [latex]\dfrac{\pi}{3}[/latex]
    2. [latex]\dfrac{7\pi}{6}[/latex]
    3. [latex]\dfrac{7\pi}{4}[/latex]

 

Activity 6.4. Graphs of Sine and Cosine.

We are going to graph [latex]f(t) = \sin t[/latex] and [latex]g(t) = \cos t[/latex] from their definitions. The unit circle [latex]x^2 + y^2 = 1[/latex] at the left of each grid is marked off in radians. (Each tick mark is [latex]0.1[/latex] radian.) The horizontal [latex]t[/latex]-axis of each grid is also marked in radians.

  1. Choose a value of [latex]t[/latex] along the horizontal axis of the [latex]f(t) = \sin t[/latex] grid. This value of [latex]t[/latex] represents an angle in radians.
  2. Now look at the unit circle and find the point [latex]P[/latex] designated by that same angle in radians.
  3. Measure the vertical (signed) distance that gives the [latex]y[/latex]-coordinate of point [latex]P{.}[/latex]
  4. At the value of [latex]t[/latex] you chose in step 1, lightly draw a vertical line segment the same length as the [latex]y[/latex]-coordinate of [latex]P{.}[/latex] Put a dot at the top (or bottom) of the line segment.
  5. Repeat for some more values of [latex]t{.}[/latex] Connect the dots to see the graph of [latex]f(t) = \sin t{.}[/latex]

circle and grid

  1. Choose a value of [latex]t[/latex] along the horizontal axis of the [latex]g(t) = \cos t[/latex] grid. This value of [latex]t[/latex] represents an angle in radians.
  2. Now look at the unit circle and find the point [latex]P[/latex] designated by that same angle in radians.
  3. Measure the horizontal (signed) distance that gives the [latex]x[/latex]-coordinate of point [latex]P{.}[/latex]
  4. At the value of [latex]t[/latex] you chose in step 1, lightly draw a vertical line segment the same length as the [latex]x[/latex]-coordinate of [latex]P{.}[/latex] Put a dot at the top (or bottom) of the line segment.
  5. Repeat for some more values of [latex]t{.}[/latex] Connect the dots to see the graph of [latex]g(t) = \cos t{.}[/latex]

unit circle and grid

 

Activity 6.5. Solving Equations.

unit circle and sine graph

      1. Use the graph of [latex]y=\sin t[/latex] to estimate two solutions of the equation [latex]\sin t = 0.65{.}[/latex] Show your solutions on the graph.
      2. Use the unit circle to estimate two solutions of the equation [latex]\sin t = 0.65{.}[/latex] Show your solutions on the circle.
      1. Use the graph of [latex]y=\sin t[/latex] to estimate two solutions of the equation [latex]\sin t = -0.2{.}[/latex] Show your solutions on the graph.
      2. Use the unit circle to estimate two solutions of the equation [latex]\sin t = -0.2{.}[/latex] Show your solutions on the circle.

unit circle and grid

      1. Use the graph of [latex]y=\cos t[/latex] to estimate two solutions of the equation [latex]\cos t = 0.15{.}[/latex] Show your solutions on the graph.
      2. Use the unit circle to estimate two solutions of the equation [latex]\cos t = 0.15{.}[/latex] Show your solutions on the circle.
        1. Use the graph of [latex]y=\cos t[/latex] to estimate two solutions of the equation [latex]\cos t = -0.4{.}[/latex] Show your solutions on the graph.
        2. Use the unit circle to estimate two solutions of the equation [latex]\cos t = -0.4{.}[/latex] Show your solutions on the circle.

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