Chapter 6 Summary and Review

Key Concepts

1. The distance we travel around a circle of radius is proportional to the angle of displacement.
   [latex]{Arclength}~ = ~ ({fraction ~~of ~~one ~~revolution}) \cdot (2\pi r)[/latex]
2. We measure angles in radians when we work with arclength.
   

   Radians.

   The radian measure of an angle is given by

   [latex]({fraction ~~of ~~one ~~revolution}\times 2\pi)[/latex]

3. An arclength equal to one radius determines a central angle of one radian.
4. Radian measure can be expressed as multiples of [latex]\pi[/latex] or as decimals.
   

   Degrees	[latex]\dfrac{{Radians:}}{{Exact ~~Values}}[/latex]	[latex]\dfrac{{Radians: Decimal}}{{Approximations}}[/latex]
   [latex]0°[/latex]	[latex]0[/latex]	[latex]0[/latex]
   [latex]90°[/latex]	[latex]\dfrac{\pi}{2}[/latex]	[latex]1.57[/latex]
   [latex]180°[/latex]	[latex]\pi[/latex]	[latex]3.14[/latex]
   [latex]270°[/latex]	[latex]\dfrac{3\pi}{2}[/latex]	[latex]4.71[/latex]
   [latex]360°[/latex]	[latex]2\pi[/latex]	[latex]6.28[/latex]

   

5. We multiply by the appropriate conversion factor to convert between degrees and radians.
   

   Unit Conversion for Angles.

   [latex]\dfrac{180°}{\pi~{radians}} = 1[/latex]

   To convert from radians to degrees we multiply the radian measure by [latex]\dfrac{180°}{\pi}{.}[/latex]

   To convert from degrees to radians we multiply the degree measure by [latex]\dfrac{\pi}{180}{.}[/latex]

6. Arclength Formula.

   On a circle of radius [latex]r{,}[/latex] the length [latex]s[/latex] of an arc spanned by an angle [latex]\theta[/latex] in radians is

   [latex]s = r\theta[/latex]

7. On a unit circle, the measure of a (positive) angle in radians is equal to the length of the arc it spans.
8. The sine, cosine, or tangent of a particular angle is the same whether the angle is measured in radians or in degrees.
9. You should memorize the trig values of the special angles in radians.
   

   Degrees	Radians	Sine	Cosine	Tangent
   [latex]0°[/latex]	[latex]0[/latex]	[latex]0[/latex]	[latex]1[/latex]	[latex]0[/latex]
   [latex]30°[/latex]	[latex]\dfrac{\pi}{6}[/latex]	[latex]\dfrac{1}{2}[/latex]	[latex]\dfrac{\sqrt{3}}{2}[/latex]	[latex]\dfrac{1}{\sqrt{3}}[/latex]
   [latex]45°[/latex]	[latex]\dfrac{\pi}{4}[/latex]	[latex]\dfrac{1}{\sqrt{2}}[/latex]	[latex]\dfrac{1}{\sqrt{2}}[/latex]	[latex]1[/latex]
   [latex]60°[/latex]	[latex]\dfrac{\pi}{3}[/latex]	[latex]\dfrac{\sqrt{3}}{2}[/latex]	[latex]\dfrac{1}{2}[/latex]	[latex]\sqrt{3}[/latex]
   [latex]90°[/latex]	[latex]\dfrac{\pi}{2}[/latex]	[latex]1[/latex]	[latex]0[/latex]	undefined

10. To find the sine or cosine of a real number [latex]t{,}[/latex] we draw an arc of length [latex]t[/latex] on a unit circle, and then find the sine or cosine of the angle [latex]\theta[/latex] determined by the arc.

11. Coordinates on a Unit Circle.

    The coordinates of the point [latex]P[/latex] determined by an arc of length [latex]t[/latex] in standard position on a unit circle are

    [latex](x, y) = (\cos t, \sin t)[/latex]

    

12. The Circular Functions.

    Let [latex]P[/latex] be the terminal point of an arc of length [latex]t[/latex] in standard position on a unit circle. The circular functions of [latex]t[/latex] are defined by

    [latex][latex] \begin{aligned}[t] \cos t = x\\ \sin t = y\\ \tan t = \dfrac{y}{x},~~x \not= 0\\ \end{aligned}[/latex]

    

13. The domain of a function is the set of all possible input values. The range of a function is the set of all output values for the function.

14. [latex]f(x) = \sin x[/latex].

    Domain: all real numbers
    Range: [latex][-1,1][/latex]

    Period: [latex]2\pi[/latex]

    

    [latex]g(x) = \cos x[/latex].

    Domain: all real numbers
    Range: [latex][-1,1][/latex]

    Period: [latex]2\pi[/latex]

    

    [latex]h(x) = \tan x[/latex].

    Domain: all real numbers except all the integer multiple of [latex]\dfrac{\pi}{2}[/latex]

    Range: all real numbers
    Period: [latex]\pi[/latex]