Basic Sine and Cosine Curves: Practice Problems

1: Graphing Sine and Cosine

First, start by graphing sine and cosine. You should be able to graph these from memory!

2: Properties of Sine and Cosine Functions

For both the sine and cosine functions, answer the following questions:

• What is the domain?
• What is the range?
• What is the period of the function?
• What are the $x$-intercepts?
• What are the $y$-intercepts?
• Is it odd or even?
• What are the line(s) of symmetry for the graph?

3: Period of Sine and Cosine Functions

Remember that:

If $b$ is a positive, real number, then the period of $y=a \sin bx$ and $y=a \cos bx$ is given by: $Period=\frac{2\pi}{b}$

Knowing this, find the PERIOD of the following:

• $y=7\sin(10x)$
• $y=2\cos(16x)$
• $y=\sin(3x)$

Now, graph the three functions above, given that you now know the periods for each.

4: Describing the Relationship Between Graphs

For the following, describe the relationship between the graphs of $f$ and $g$. Consider amplitudes, periods, and shifts.

a

$f(x)=\sin x$ $g(x)=\sin(x-\pi)$

b

$f(x)=\cos 2x$ $g(x)=-\cos 2x$

c

$f(x)=\cos 2x$ $g(x)=3+\cos 2x$

d

$f(x)=\sin 3x$ $g(x)=\sin(-3x)$

e

$f(x)=\sin x$ $g(x)=-\frac{1}{2} \sin x$

f

$f(x)=\cos 4x$ $g(x)=-2+\cos 4x$

5: Sketching the Graphs of Sine or Cosine Functions

For the following, sketch the graphs of $f$ and $g$ in the same coordinate plane. Include two full periods.

a

$f(x)=\sin x$ $g(x)=-4 \sin x$

b

$f(x)= \cos \pi x$ $g(x)=1 + \cos \pi x$

c

$f(x)=-\frac{1}{2} \sin \frac{x}{2}$ $g(x)=2 \sin(\frac{x}{4})$

6: Tangent Function

For both the tangent, answer the following questions:

• What is the domain?
• What is the range?
• What is the period of the function?
• What are the $x$-intercepts?
• What are the $y$-intercepts?
• Is it odd or even?
• What are the line(s) of symmetry for the graph?

7: Graphing Other Trig Functions

Graph the functions for:

• Cotangent
• Cosecant
• Secant

If you want, answer these questions for these three as well:

• What is the domain?
• What is the range?
• What is the period of the function?
• What are the $x$-intercepts?
• What are the $y$-intercepts?
• Is it odd or even?
• What are the line(s) of symmetry for the graph?