27.02 Graphs of sine and cosine functions (Extended)
The value of \sin \theta can be represented on the given xy-plane.
Consider the curve y = \sin \theta:
Complete the following:
A rotation of 380 \degree has the same rotation as the acute angle with measure ⬚. So \sin 380 \degree = \sin ⬚.
A rotation of 480 \degree has the same rotation as the obtuse angle with measure ⬚. So \sin 480 \degree = \sin ⬚.
A rotation of 580 \degree has the same rotation as the reflex angle with measure ⬚. So \sin 580 \degree = \sin ⬚ \degree.
Consider the graph of y = \sin \theta for \\ 0 \leq \theta \leq 720 \degree:
Describe the nature of y = \sin \theta.
The number of degrees it takes for the curve to complete a full cycle is called the period of the function.
Determine the period of y = \sin \theta.
Consider the curve y = \sin x and state whether the following statements are true or false:
The graph of y = \sin x is periodic.
As x approaches infinity, the height of the graph approaches infinity.
The graph of y = \sin x is increasing between x = \left( - 90 \right) \degree and x = 0 \degree.
The graph of y = \sin x is symmetric about the line x = 0.
The graph of y = \sin x is symmetric with respect to the origin.
The y-values of the graph repeat after a period of 360 \degree.
Consider the curve y = \cos x and state whether the following statements are true or false:
The graph of y = \cos x is cyclic.
As x approaches infinity, the height of the graph approaches infinity.
The graph of y = \cos x is increasing between x = \left( - 180 \right) \degree and x = \left( - 90 \right) \degree.
The graph of y = \cos x has reflective symmetry across the line x = 0.
The graph of y = \cos xhas rotational symmetric with respect to the origin.
The y-values of the graph repeat after a period of 180 \degree.
Consider the curve y = \sin x:
State the coordinates of the y-intercept.
Find the maximum y-value.
Find the minimum y-value.
Consider the curve y = \cos x:
State the coordinates of the y-intercept.
Find the maximum y-value.
Find the minimum y-value.
Consider the equation y = \cos x.
Complete the table of values:
| x | 0 \degree | 90 \degree | 180 \degree | 270 \degree | 360 \degree |
|---|---|---|---|---|---|
| \cos x |
Sketch the graph of y = \cos x.
Consider the equation y = \sin x.
Complete the table of values:
| x | 0 \degree | 90 \degree | 180 \degree | 270 \degree | 360 \degree |
|---|---|---|---|---|---|
| \sin x |
Sketch the graph of y = \sin x.
Consider the graph of y = \cos x:
Is \cos 162 \degree positive or negative?
In which quadrant does an angle with measure 162 \degree lie in?
Consider the graph of y = \sin x:
Is \sin 200 \degree positive or negative?
In which quadrant does an angle with measure 200 \degree lie in?
Consider the curve y = \sin x:
If one cycle of the graph of y = \sin x starts at x = 0 \degree, when does the next cycle start?
Find the x-value of the x-intercept in the following regions:
0 \degree < x < 360 \degree
\left( - 360 \right) \degree < x < 0 \degree
True or false: As x approaches infinity, the graph of y = \sin x stays between y = - 1 and y = 1.
State whether the graph of y = \sin x increases or decreases in the following regions:
90 \degree < x < 270 \degree
\left( - 270 \right) \degree < x < \left( - 90 \right) \degree
\left( - 450 \right) \degree < x < \left( - 270 \right) \degree
\left( - 90 \right) \degree < x < 90 \degree
270 \degree < x < 450 \degree
Consider the curve y = \cos x:
If one cycle of the graph of y = \cos x starts at x = \left( - 90 \right) \degree, when does the next cycle start?
Find the x-values of the x-intercepts in the following regions:
0 \degree < x < 360 \degree
\left( - 360 \right) \degree < x < 0 \degree
True or false: As x approaches infinity, the graph of y = \cos x stays between y = - 1 and y = 1.
State whether the graph of y = \cos x increases or decreases in the following regions:
0 \degree < x < 180 \degree
\left( - 180 \right) \degree < x < 0 \degree
180 \degree < x < 360 \degree
\left( - 360 \right) \degree < x < \left( - 180 \right) \degree