##### Notes

This printable supports Common Core Mathematics Standard HSF-TF.A.4

##### Print Instructions

NOTE: Only your test content will print.
To preview this test, click on the File menu and select Print Preview.

See our guide on How To Change Browser Print Settings to customize headers and footers before printing.

# Periodicity of Trig Functions (Grades 11-12)

Print Test (Only the test content will print)

## Periodicity of Trig Functions

1.
Which of the following are equal to $sin(pi/4) ?$ Choose all that apply.
1. $sin((-7pi)/4)$
2. $sin((17pi)/4)$
3. $sin((13pi)/4)$
4. $sin((5pi)/4)$
2.
Which of the following are equal to $cos(pi/12) ?$ Choose all that apply.
1. $cos((-35pi)/12)$
2. $cos((-pi)/12)$
3. $cos((25pi)/12)$
4. $cos((97pi)/12)$
3.
Which of the following are equal to $tan((5pi)/7) ?$ Choose all that apply.
1. $tan((2pi)/7)$
2. $tan((-2pi)/7)$
3. $tan((12pi)/7)$
4. $tan((18pi)/7)$
4.
Which of the following are equal to $sin((7pi)/3) ?$ Choose all that apply.
1. $sin((4pi)/3)$
2. $sin(pi/3)$
3. $sin((-5pi)/3)$
4. $sin((-17pi)/3)$
5.
Which of the following are equal to $cos((4pi)/9) ?$ Choose all that apply.
1. $cos((-5pi)/9)$
2. $cos((13pi)/9)$
3. $cos((22pi)/9)$
4. $cos((-23pi)/9)$
6.
Which of the following are equal to $tan((13pi)/3) ?$ Choose all that apply.
1. $tan((10pi)/3)$
2. $tan((2pi)/3)$
3. $tan((-pi)/3)$
4. $tan((-11pi)/3)$
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 $T!=0$, $f(x+T)=f(x)$ for all $x$ in the domain of $f$.

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