Graphing trig functions without calculus

[nietzsche]

Homework Statement

I'm in a first-year analysis course, and this question was given by my prof. as practice for her midterm test.

"Sketch the graph of the function

$$\begin{equation*} f(x) = \text{sin} 2x + \sqrt{3} \text{cos} 2x \end{equation*}$$

Determine the amplitude, the frequency and the phase of $$f(x)$$.

Homework Equations

The Attempt at a Solution

It's been a really long time since I've done graphing transformations of trig functions without calculus, so I truthfully don't remember how to do this.

I can see they both have period pi, and sin(2x) has amplitude 1 and sqrt3 cos(2x) has amplitude of sqrt3.

I don't know how to determine the highest point on the graph (amplitude), and I don't know how to determine the phase.

Thanks for your help.

 

[Dick]

http://en.wikipedia.org/wiki/List_of_trigonometric_identities Look at the section called "Linear Combinations".

 

[nietzsche]

http://en.wikipedia.org/wiki/List_of_trigonometric_identities Look at the section called "Linear Combinations".

wow, thanks. i don't think I've actually ever learned this. i'll have to look over it.

 

[nietzsche]

i found this

http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/web-rcostheta-alphaetc.pdf

which explains it in terms of a trig identity.

the answer i got is

$$2\text{cos}(2x-\dfrac{\pi}{6})$$

So amplitude is 2, frequency is 1/pi, and it has a phase shift of pi/12 to the right.

 

[Dick]

i found this

http://www.mathcentre.ac.uk/resources/workbooks/mathcentre/web-rcostheta-alphaetc.pdf

which explains it in terms of a trig identity.

That's a nice detailed exposition.