Trigonometric Half Angle Formula Explained and Applied

[jrjack]

Homework Statement

$$\tan[\frac{1}{2} \arcsin(\frac{-7}{25})]$$

The Attempt at a Solution

I'm not sure how to take 1/2 the arcsin, should this use the half-angle formula?

Normally I would draw a triangle using the sin value (-7/25), then find the tan value (24/25), but the 1/2 is throwing me off.

How do I start this? Is this 1/2 the sin value (-7/25)= -7/50, then solve for the tan(-7/50)?

 

[Dick]

Use the tangent half angle formula. tan(x/2)=??

 

[jrjack]

Thanks, so i get

$$-\sqrt{26}$$

Does that sound right?

 

[Dick]

Thanks, so i get

$$-\sqrt{26}$$

Does that sound right?

arcsin(-7/25) is about -0.3. If you take half of that and take the tangent, it's nowhere near -sqrt(26) which is about -5. Is it? You can check these solutions using rough estimates or a calculator.

 

[jrjack]

$$-\sqrt{\frac{1+\cos x}{1-\cos x}}$$
$$-\sqrt{\frac{1+\frac{24}{25}}{1-\frac{24}{25}}}$$
$$=-\sqrt{26}$$

 

[Dick]

$$-\sqrt{\frac{1+\cos x}{1-\cos x}}$$
$$-\sqrt{\frac{1+\frac{24}{25}}{1-\frac{24}{25}}}$$
$$=-\sqrt{26}$$

tan(0/2)=0. If you put x=0 into your supposed half angle formula, what do you get? Does it work?

 

[jrjack]

Sorry, I now realize I have my signs flipped in my formula.
I think my answer should be:$$-\sqrt{\frac{1}{26}}$$

 

[Dick]

Sorry, I now realize I have my signs flipped in my formula.
I think my answer should be:$$-\sqrt{\frac{1}{26}}$$

That doesn't work either because (1-24/25)/(1+24/25) isn't equal to 1/26. Now what's it really equal to??

 

[jrjack]

Sorry, I got in a hurry, between typing the tex and working the problem several different ways (wrong of course).

It should equal 1/49, which means my answer should be $-\sqrt{\frac{1}{49}}$

 

[Dick]

Sorry, I got in a hurry, between typing the tex and working the problem several different ways (wrong of course).

It should equal 1/49, which means my answer should be $-\sqrt{\frac{1}{49}}$

Ok, aside from the fact there is a simpler way to write -1/sqrt(49) could you try and check that using a calculator from your original expression? It's really useful to have a simple way of self-checking whether you are way off or not.

 

[jrjack]

Thank you for your help.
I realize I still need to rationalize the denominator, and after checking with my calculator both answers come out to -.142857, so that must be correct.

My final answer should be -1/7

Once again, thank you for your help.

 

[Dick]

Thank you for your help.
I realize I still need to rationalize the denominator, and after checking with my calculator both answers come out to -.142857, so that must be correct.

My final answer should be -1/7

Once again, thank you for your help.

Very welcome and quite right. The main lesson is how easy these answers are to check with a calculator.