Periodicity of Trig Functions (Grades 11-12)

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Periodicity of Trig Functions

1. 
Which of the following are equal to [math]sin(pi/4) ?[/math] Choose all that apply.
  1. [math]sin((-7pi)/4)[/math]
  2. [math]sin((17pi)/4)[/math]
  3. [math]sin((13pi)/4)[/math]
  4. [math]sin((5pi)/4)[/math]
2. 
Which of the following are equal to [math]cos(pi/12) ?[/math] Choose all that apply.
  1. [math]cos((-35pi)/12)[/math]
  2. [math]cos((-pi)/12)[/math]
  3. [math]cos((25pi)/12)[/math]
  4. [math]cos((97pi)/12)[/math]
3. 
Which of the following are equal to [math]tan((5pi)/7) ?[/math] Choose all that apply.
  1. [math]tan((2pi)/7)[/math]
  2. [math]tan((-2pi)/7)[/math]
  3. [math]tan((12pi)/7)[/math]
  4. [math]tan((18pi)/7)[/math]
4. 
Which of the following are equal to [math]sin((7pi)/3) ?[/math] Choose all that apply.
  1. [math]sin((4pi)/3)[/math]
  2. [math]sin(pi/3)[/math]
  3. [math]sin((-5pi)/3)[/math]
  4. [math]sin((-17pi)/3)[/math]
5. 
Which of the following are equal to [math]cos((4pi)/9) ?[/math] Choose all that apply.
  1. [math]cos((-5pi)/9)[/math]
  2. [math]cos((13pi)/9)[/math]
  3. [math]cos((22pi)/9)[/math]
  4. [math]cos((-23pi)/9)[/math]
6. 
Which of the following are equal to [math]tan((13pi)/3) ?[/math] Choose all that apply.
  1. [math]tan((10pi)/3)[/math]
  2. [math]tan((2pi)/3)[/math]
  3. [math]tan((-pi)/3)[/math]
  4. [math]tan((-11pi)/3)[/math]
7. 
The following questions will investigate the period of the sine, cosine, and tangent functions. The definition of a periodic function is that, for some constant [math]T!=0[/math], [math]f(x+T)=f(x)[/math] for all [math]x[/math] in the domain of [math]f[/math].

For the following questions, let [math]theta[/math] be an angle in standard position. Let the intersection of the terminal arm of [math]theta[/math] and the unit circle centered at the origin be [math](a,b)[/math].
A. 
Since the sine function is periodic, we know from the definition of periodic functions that [math]sin(theta + T) = sin(theta)[/math] for some value [math]T!=0[/math]. Since this must be true for all values of [math]theta[/math], one can look at the specific value [math]theta=0[/math]. Then [math]sin(T) = sin(0) = 0[/math]. Using the unit circle, what values of [math]T[/math], other than zero, satisfy this equation?
  1. Any integer multiple of [math]pi/2[/math].
  2. Any integer multiple of [math]pi[/math].
  3. Any integer multiple of [math]2pi[/math].
  4. The values [math]pi[/math] and [math]2pi[/math].
B. 
Which of the following gives a valid counterexample, showing that [math]T=pi[/math] is not the period of the sine function, but only a special case when [math]theta=0 ?[/math] Choose all that apply.
  1. [math]theta=pi/3[/math]
  2. [math]theta = -pi[/math]
  3. [math]theta =2pi[/math]
  4. [math]theta =pi/2[/math]
C. 
The period of a periodic function is the smallest value of [math]T[/math] such that [math]f(x+T) = f(x)[/math] for all values of [math]x[/math] in the domain of [math]f[/math]. Using the information in the previous questions, which of the following gives the best reasoning as to why the period of the sine function is [math]2pi ?[/math]
  1. Since values of [math]T[/math] less than [math]2pi[/math] weren't valid, [math]2pi[/math] must work.
  2. Using [math]T=2pi[/math] and trying a few values of [math]theta[/math], such as [math](3pi)/7[/math] and [math](8pi)/11[/math], the equation [math]sin(theta+2pi) = sin(theta)[/math] is valid. Therefore, it is true for all values of [math]theta[/math].
  3. For [math]T=2pi[/math], [math]theta + T[/math] and [math]theta[/math] are coterminal angles for any value of [math]theta[/math]. Therefore, referring to the unit circle, [math]sin(theta)=b[/math] and [math]sin(theta+2pi)=b[/math], and therefore [math]sin(theta+2pi)=sin(theta)[/math].
  4. There are no other values of [math]T[/math] such that [math]sin(x+T) = sin(x)[/math].
D. 
The period of the cosine function is also [math]2pi[/math], and showing this is true is similar to the process used for the sine function. However, the period of the tangent function is [math]pi[/math]. Which of the following equations correctly shows why the period of tangent is [math]pi?[/math]
  1. [math]tan(theta + pi) = sin(theta+pi)/cos(theta+pi) = (-sin(theta))/(-cos(theta)) = sin(theta)/cos(theta) = tan(theta)[/math]
  2. [math]tan(theta + pi) = sin(theta+pi)/cos(theta+pi) = (sin(theta)+pi)/(cos(theta)+pi) = sin(theta)/cos(theta) = tan(theta)[/math]
  3. [math]tan(theta + pi) = sin(theta+pi)/cos(theta+pi) = (sin(theta)+sin(pi))/(cos(theta)+cos(pi)) = (sin(theta)+0)/(cos(theta)+0) = tan(theta)[/math]
  4. [math]tan(theta + pi) = tan(theta) + tan(pi) = tan(theta)[/math]

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