# Homework Help: Question about sin rules

1. Oct 8, 2012

### hahaha158

1. The problem statement, all variables and given/known data
I have the equation (sin(2x))/x = ?

3. The attempt at a solution

I know that the answer to this is 2, but im not sure why (sin(2x))/x = 2

Can somone explain?

Thanks

2. Oct 8, 2012

### Staff: Mentor

(sin(2x))/x ≠ 2, so perhaps you are leaving something out of the problem. What is the complete problem statement?

3. Oct 8, 2012

### hahaha158

Find the value of the constant a for which the function below is continuous everywhere. Fully

........... a+x2 while x≤0
f(x) = {
........... (sin(2x))/x while x>0

4. Oct 8, 2012

### Staff: Mentor

The problem is really asking about limits, namely
$$\lim_{x \to 0}\frac{sin(2x)}{x}$$

Do you know any other limits that involve trig functions?

5. Oct 8, 2012

### hahaha158

I'm not sure what you mean by your question.

I know that $$\lim_{x \to 0}\frac{sin(x)}{x}$$ is equal to 1

I also know that the limit as x approaches 0 from the negative and the limit at x=0 are both just equal to a.

So this means that a just equals $$\lim_{x \to 0}\frac{sin(2x)}{x}$$

I know the answer to the question is 2 so that means $$\lim_{x \to 0}\frac{sin(2x)}{x}$$ must equal 2 but im not sure how to do it.

6. Oct 8, 2012

### Staff: Mentor

There are at least a couple of ways to go.
1) Double angle identity for sine
2) Adjust things so that you have sin(2x)/(2x) times some other stuff.

7. Oct 8, 2012

### hahaha158

For 2) do you mean like

(sin(2x))/2x)*2

=so you get 1*2

=2?

Would that work?

8. Oct 8, 2012

### 5ymmetrica1

(sin(2x))/ x = (sin2(1))/ 1
= sin(2)(1)/1
= sin(2)
= 0.0349

9. Oct 8, 2012

### Mentallic

That's the limit as x goes to 1, not 0.

Yep, exactly! This would be the simplest way to do it, so if you ever get a question like

$$\lim_{x\to 0}\frac{\sin(ax)}{b}$$ then this is equivalent to
$$\lim_{x\to 0}\frac{a}{b}\cdot\frac{\sin(ax)}{a}=\frac{a}{b}$$