## anonymous one year ago what are the steps to solve this problem? Write each trigonometric expression as an algebraic expression in x 1)sin(tan^-1x) 2)tan(cos^-1x)

1. jim_thompson5910

Draw a right triangle |dw:1436829693684:dw|

2. jim_thompson5910

If we let $$\Large \theta = \tan^{-1}(x)$$ then $$\Large \tan(\theta) = x = \frac{x}{1}$$ because tangent = opposite/adjacent, we can add these labels |dw:1436829773980:dw| making sense so far?

3. anonymous

yup :)

4. jim_thompson5910

what is the length of the hypotenuse?

5. anonymous

r=1^2+x^2 so x^2?

6. jim_thompson5910

close, but the square root of 1^2+x^2 or the square root of 1+x^2 |dw:1436830005407:dw|

7. anonymous

oh okay

8. jim_thompson5910

with that triangle, you can find sin(theta) remember I made theta be equal to the inverse tangent of x

9. anonymous

cos=y/r 1/sqrtx^2+1 1/x+1

10. anonymous

hmm I think I did something wrong

11. jim_thompson5910

you want sine, so sine = opposite/hypotenuse $\Large \sin(\theta) = \frac{x}{\sqrt{x^2+1}}$ $\Large \sin\left(\tan^{-1}(x)\right) = \frac{x}{\sqrt{x^2+1}}$

12. jim_thompson5910

|dw:1436830701065:dw|

13. jim_thompson5910

btw, $\Large \sqrt{x^2+1} \ne x+1$ $\Large \sqrt{x^2+y^2} \ne x+y$

14. jim_thompson5910

and $\Large \sqrt{A+B} \ne \sqrt{A} + \sqrt{B}$

15. anonymous

oh sorry I didn't know that :o

16. anonymous

how would I simplify it?

17. jim_thompson5910

you can rationalize the denominator, but you can't really do anything else to simplify

18. anonymous

I would rationalize it by multiplying both sides by sqrtx^2+1 right?

19. jim_thompson5910

top and bottom by sqrt(x^2+1)

20. anonymous

ohh okay I understand the problem now :)

21. jim_thompson5910

Rationalizing the denominator gives... $\Large \sin\left(\tan^{-1}(x)\right) = \frac{x}{\sqrt{x^2+1}}$ $\Large \sin\left(\tan^{-1}(x)\right) = \frac{x{\color{red}{*\sqrt{x^2+1}}}}{\sqrt{x^2+1}{\color{red}{*\sqrt{x^2+1}}}}$ $\Large \sin\left(\tan^{-1}(x)\right) = \frac{x\sqrt{x^2+1}}{(\sqrt{x^2+1})^2}$ $\Large \sin\left(\tan^{-1}(x)\right) = \frac{x\sqrt{x^2+1}}{x^2+1}$ to me, that's not really simplified. If anything, it got more complicated. But some books require you to rationalize the denominator

22. anonymous

I see

23. anonymous

thank for all your help! I get it now :D sorry it took so much time

24. jim_thompson5910

that's ok and you're welcome