# How is $\tan ( \arcsin x) = \frac{x}{\sqrt{1-x^2}}$ ?

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1. Aug 17, 2016

### Rectifier

• Member warned about posting without the template and with no effort shown
How can I write $\tan ( \arcsin x)$ as $\frac{x}{\sqrt{1-x^2}}$? This is not a problem in itself but a part of a solution to a problem.

2. Aug 17, 2016

### Staff: Mentor

Start by using the sin / cos definition of tan and you should begin to see the derivation.

3. Aug 17, 2016

### Rectifier

$\tan ( \arcsin x) = \frac{\sin ( \arcsin x)}{\cos ( \arcsin x)} = \frac{x}{\cos ( \arcsin x)}$

I draw a triangle with sides x and 1. Therefore the hypotenuse is $\sqrt{1+x^2}$.

$\cos \left( v \right) = \frac{1}{\sqrt{1+x^2}}$

4. Aug 17, 2016

### Staff: Mentor

If $\theta=arcsin x$, what is $\sin \theta?$

5. Aug 17, 2016

### Mastermind01

You drew the wrong triangle. You need a triangle with an angle whose sine is x. Then which of the sides needs to be x and 1?

6. Aug 17, 2016

### Rectifier

If $\theta = \arcsin (\sin x)$

$x$ is the opposite side of the angle $\theta$ and 1 is the hypotenuse?.

7. Aug 17, 2016

### Mastermind01

Does that get you the answer?

8. Aug 17, 2016

### Staff: Mentor

No. What is the sine of the angle whose sine is x?

9. Aug 17, 2016

### Mastermind01

I think he got it correct this time.

10. Aug 17, 2016

### Rectifier

Hmm....
$\theta = \arcsin (\sin x)$

Then $\sin \theta = \sin x$

But this looks wrong

Sorry, this is hard for me :S

11. Aug 17, 2016

### Mastermind01

Well you drew the triangle correct, so what's $cos(\theta)$ ?

12. Aug 17, 2016

### Rectifier

Wait, so one side of the triangle is equal to the angle?

$\cos x = \sqrt{1-x^2}$

13. Aug 17, 2016

### Mastermind01

So $tan(\theta)$ is?

14. Aug 17, 2016

### Rectifier

$\frac{x}{\sqrt{1-x^2}}$ :D

15. Aug 17, 2016

### Staff: Mentor

No. $sin \theta = x$

16. Aug 17, 2016

### Staff: Mentor

In post #8, I asked you "What is the sine of the angle whose sine is x?" You answered sin x. What would your answer have been if I asked "What is the name of the person whose name is John?"

17. Aug 17, 2016

### Rectifier

It is x :)

18. Aug 17, 2016

### Staff: Mentor

Good. So, in terms of x, what is cos theta? Then , in terms of x, what is tan theta?