Solving Trigonometric Equations Using Identities

[Elijah the Wood]

1)
tan(x)-sin(x)/2tan(x) = sin^2(x/2)
1-sin(x)/2 = sin^2(x/2)

(That's as far as I've gotten on this one...but i don't know if I'm in the right direction or what to do next?)

2)
sin2x = 1 / tanx + cot2x
2sin(x)cos(x) = ''
sin(x) * cos(x) * sin(x) = ''

(The same from #1 applies to this problem)

 

[Pyrrhus]

Assuming you mean:
$$\frac{\tan(x)-\sin(x)}{2 \tan(x)} = \sin^{2}(\frac{x}{2})$$

Let's change everything to cosinus and sinus

$$\frac{\frac{\sin(x)}{\cos(x)} - \sin(x)}{\frac{2 \sin(x)} {\cos(x)}} = \sin^{2}(\frac{x}{2})$$

Simplifying:

$$\frac{\frac{\sin(x) - \sin(x) \cos(x)}{\cos(x)}}{\frac{2 \sin(x)} {\cos(x)}} = \sin^{2}(\frac{x}{2})$$

$$\frac{\sin(x) - \sin(x) \cos(x)}{2 \sin(x)} = \sin^{2}(\frac{x}{2})$$

$$\frac{1 - \cos(x)}{2} = \sin^{2}(\frac{x}{2})$$

Remember $\sin^{2}(\frac{x}{2}) = \frac{1 - \cos(x)}{2}$

$$\frac{1 - \cos(x)}{2} = \sin^{2}(\frac{x}{2})$$

$$\sin^{2}(\frac{x}{2}) = \sin^{2}(\frac{x}{2})$$

Assuming you mean:
$$\sin (2x) = \frac{1}{\tan(x) + \cot(2x)}$$

Working the right to the left

Changing to sinus and cosinus:

$$\sin (2x) = \frac{1}{\frac{\sin(x)}{\cos(x)} + \frac{\cos(2x)}{\sin(2x)}}$$

Applying double angle identities

$$\sin (2x) = \frac{1}{\frac{\sin(x)}{\cos(x)} + \frac{\cos^2(x) - \sin^2(x)}{2 \sin(x) \cos(x)}}$$

Simplifying:

$$\sin (2x) = \frac{1}{\frac{2 \sin^2(x) \cos(x) + \cos^3(x) - \sin^2(x) \cos(x)}{2 \sin(x) \cos^2(x)}}$$

$$\sin (2x) = \frac{1}{\frac{\sin^2(x) \cos(x) + \cos^3(x)}{2 \sin(x) \cos^2(x)}}$$

$$\sin (2x) = \frac{1}{\frac{\sin^2(x) + \cos^2(x)}{2 \sin(x) \cos(x)}}$$

Remember $\sin^2(x) + \cos^2(x) = 1$

$$\sin (2x) = \frac{1}{\frac{1}{2 \sin(x) \cos(x)}}$$

Remember $\sin(2x) = 2 \sin(x) \cos(x)$

$$\sin (2x) = 2 \sin(x) \cos(x)$$

$$\sin (2x) = \sin (2x)$$

 

[jai6638]

damn it... how do u do all these equations on the computer?? i find it hard doing my math on a computer... I am more comfortable with a pen and paper... get tired of using the "^",etc

any words of wisdom ?

 

[Elijah the Wood]

Jeez...that was incredible. Thanks a lot...but I don't suppose there was an easier way, was there? haha

 

[Pyrrhus]

damn it... how do u do all these equations on the computer?? i find it hard doing my math on a computer... I am more comfortable with a pen and paper... get tired of using the "^",etc

any words of wisdom ?

You get used to imagining it in your head.

 

[jai6638]

lol i imagine it in my head too when possible but i just can't get myself to type it all down...Good ol' pen and paper!