# Thread: Understanding the Unit circle and the Trigonometric functions

1. ## Understanding the Unit circle and the Trigonometric functions

Dear members of the Math Help Forum,

I am going to be soon entering high school(the Finnish school system), and thus will be taking my first Calculus course. During the summer I decided to look at some pre-calculus topics, which are going to be quickly covered before moving on to calculus. Well, guess what? Two months hve gone by and I am still stuck on trigonometry.
When we studied trigonometry in comprehensive school(the right-angled triangles), I understood it fine, but as soon as my Calculus book started talking about the Unit circle and the cosine, sine and tangent functions, I just couldn't understand it anymore. So to this day I still am stuck on trigonometry.

For example, the exercise in my book asks me to prove cos^4x-sin^4x=cos(2x)

I was able to prove one of the addition formulas cos(s-t)=cos s cos t + sin s sin t, because the book provided a visual representation of what this formula is actually saying. However, for the formula above I just cannot draw any kind of representation.

I have tried to search for numerous solutions to my problem, but every book I read on the topic just seems to complicate it more.

If someone could please, explain to me the concept of Unit circle, and the basic idea of the trigonometric functions, and what they are used for in Calculus or refer me to a website, which explains the concept simply, I woul be very grateful.

2. Thank you!

I have searched Dr. Math before, don't know how I missed that helpful letter.

Could someone please still help to prove the equation cos^4x - sin^4x= cos(2x) ? Thank you.

And still one more stupid question I have to ask...

Does cos(2x) mean cos(x+x) or cosx+cos x or are these two the same thing?

3. Originally Posted by Coach
Thank you!

I have searched Dr. Math before, don't know how I missed that helpful letter.

Could someone please still help to prove the equation cos^4x - sin^4x= cos(2x) ? Thank you.

And still one more stupid question I have to ask...

Does cos(2x) mean cos(x+x) or cosx+cos x or are these two the same thing?
One form for the cosine of a double angle is
$cos(2x) = cos^2(x) - sin^2(x)$

So
$cos^4(x) - sin^4(x) = (cos^2(x) - sin^2(x))(cos^2(x) + sin^2(x))$ <-- By factoring

$= cos(2x) \cdot ( 1 ) = cos(2x)$

Note: $cos(2x) \neq cos(x) + cos(x)$ except for certain x values. It is not true in general. (This means that cosine is not a linear function, as you can easily tell by looking at a graph of it.)

-Dan

4. Thank you for your answers. That helps me understand, and yet that confuses me even more.

1) I have come across another problem, and I just do not know what to do with it. Can comeone please tell me the steps they take when they solve this kind of problem?

1-cos x/sinx = sin x/ 1+ cos x = tan x/2

Thank you!

5. Use the double angle formula for $\sin x$ and $\cos x$
$\sin x=2\sin\frac{x}{2}\cos\frac{x}{2}$
$\cos x=\cos^2\frac{x}{2}-\sin^2\frac{x}{2}=2\cos^2\frac{x}{2}-1=1-2\sin^2\frac{x}{2}$