21.03 Period of sine and cosine functions
Consider the functions f \left( x \right) = \cos x and g \left( x \right) = \cos 4 x.
State the period of f \left( x \right).
Complete the table of values for g \left( x \right).
| x | 0 | 22.5 | 45 | 67.5 | 90 | 112.5 | 135 | 157.5 | 180 |
|---|---|---|---|---|---|---|---|---|---|
| g(x) |
State the period of g \left( x \right).
Describe the transformation required to obtain the graph of g \left( x \right) from f \left( x \right).
Use technology to sketch the graph of g \left( x \right) for 0 \leq x \leq 180.
Consider the function y = 2 \cos 3 x.
State the amplitude of the function.
Find the period of the function.
Use technology to sketch a graph of the function for -180 \leq x \leq 180.
For each of the following functions:
State the amplitude.
Find the period.
Use technology to sketch a graph of the function for 0 \leq x \leq 360.
Consider the function y = \sin \left( \dfrac{2x}{3} \right).
State the amplitude of the function.
Find the period of the function in degrees.
Use technology to sketch a graph of the function for 0\degree \leq x \leq 720 \degree.
A table of values for the the first period of the graph y=\sin x for x \geq 0 is given in the first table on the right:
Complete the second table given with equivalent values for x in the the first period of the graph y = \sin \left(\dfrac{x}{4}\right) for \\x \geq 0.
Hence, state the period of y = \sin \left(\dfrac{x}{4}\right).
| x | 0 | 90 | 180 | 270 | 360 |
|---|---|---|---|---|---|
| \sin x | 0 | 1 | 0 | -1 | 0 |
| x | |||||
|---|---|---|---|---|---|
| \sin\left(\dfrac{x}{4}\right) | 0 | 1 | 0 | -1 | 0 |
The functions f \left( x \right) and g \left( x \right) = f \left( kx \right) have been graphed on the same set of axes below.
Describe the transformation required to obtain the graph of g\left(x\right) from the graph of f \left( x \right).
Find the value of k.
For each of the following graphs, find the equation of the function given that it is of the form y = a \sin b x or y = a \cos b x, where b is positive:
State whether the following functions represent a change in the period from the function y = \sin x:
y = \sin \left( 5 x\right)
y = 5 \sin x
y = \sin \left( \dfrac{x}{5} \right)
y = \sin x + 5
Consider the function y = \cos 3 x + 2.
Find the period of the function.
State the amplitude of the function.
Find the maximum value of the function.
Find the minimum value of the function.
Use technology to sketch a graph of the function for 0 \leq x \leq 360.
For each of the following functions:
State the domain of the function.
State the range of the function.
Use technology to sketch a graph of the function for -180 \leq x \leq 180.
y = \sin 2 x - 2
y = - 5 \sin 2 x
y = \sin \left(\dfrac{x}{3}\right) + 5
y = \cos \left(\dfrac{x}{2}\right) - 3