Trig problem solving question

**iMan_07** · December 28th 2008, 02:19 PM

1. Given that $Sin(x) = Cos({3 \pi}{/11})$ , and x lies in the second quadrant, determine the measure of x.

I would really appreciate if someone could show me how to solve this question. Thanks.

**masters** · December 28th 2008, 02:52 PM

Originally Posted by iMan_07

I would really appreciate if someone could show me how to solve this question. Thanks.

Cosine of an angle = sine of its complement

$\cos \frac{3 \pi}{11}=\sin \left( \frac{\pi}{2}-\frac{3 \pi}{11}\right)=\sin \frac{5 \pi}{22}$

That's a Q1 angle and you want the angle in QII.

Sin is positive in Q1 and QII, so find the reference angle in QII.

$x=\pi-\frac{5 \pi}{22}=\frac{17 \pi}{22}$

**iMan_07** · December 28th 2008, 05:45 PM

Just one more quick question.....

been trying for a while now and still cant figure it out.

if sin(x)= 3/5

find tan(x/2)

using the ratios i found that tanx=3/4

but i dont know wht to do from there.

any help would be appreciated.

**Jhevon** · December 28th 2008, 05:52 PM

Originally Posted by iMan_07

Just one more quick question.....

been trying for a while now and still cant figure it out.

using the ratios i found that tanx=3/4

but i dont know wht to do from there.

any help would be appreciated.

$\tan \frac x2 = \pm \sqrt{\frac {1 - \cos x}{1 + \cos x}}$

of course, the quadrant the angle is in determines whether the answer is + or -

**Chop Suey** · December 28th 2008, 05:55 PM

Use the identity:
$\tan{\left(\frac{x}{2}\right)} = \frac{\sin{x}}{1+\cos{x}}$

Given sine, find cosine, and use the identity to find tan(x/2).

**Jhevon** · December 28th 2008, 05:56 PM

Originally Posted by Chop Suey

Use the identity:
$\tan{\left(\frac{x}{2}\right)} = \frac{\sin{x}}{1+\cos{x}}$

Given sine, find cosine, and use the identity to find tan(x/2).

nice identity

**Chop Suey** · December 28th 2008, 06:01 PM

Originally Posted by Jhevon

nice identity

You can take your identity further and multiply either the numerator or the denominator by its respective conjugate and get the identity I mentioned.

**iMan_07** · December 28th 2008, 06:03 PM

Originally Posted by Jhevon

$\tan \frac x2 = \pm \sqrt{\frac {1 - \cos x}{1 + \cos x}}$

of course, the quadrant the angle is in determines whether the answer is + or -

is there a way to solve using the double angle proof?

ie. tan2x?

**Jhevon** · December 28th 2008, 06:05 PM

Originally Posted by iMan_07

is there a way to solve using the double angle proof?

ie. tan2x?

i would not recommend it. using the half-angle formula is the simplest way i think

**iMan_07** · December 28th 2008, 06:10 PM

Originally Posted by Jhevon

i would not recommend it. using the half-angle formula is the simplest way i think

i understand that it is easier.....but we never covered half-angle and we're apparently not allowed to use it...double angle on the other hand, we used.....but i cant find a way to apply it to this question.

**Jhevon** · December 28th 2008, 06:17 PM

Originally Posted by iMan_07

i understand that it is easier.....but we never covered half-angle and we're apparently not allowed to use it...double angle on the other hand, we used.....but i cant find a way to apply it to this question.

well, if you're a glutton for punishment, you can always do

$\tan \frac x2 = \frac {\sin \frac x2}{\cos \frac x2}$

now, each of $\sin \frac x2$ and $\cos \frac x2$ can be derived using the double angle formula for $\cos 2x$

or even easier: recall that $\tan 2x = \frac {2 \tan x}{1 - \tan^2 x}$ . now replace $x$ with $\frac x2$ everywhere and solve for $\tan \frac x2$

**Chop Suey** · December 28th 2008, 06:19 PM

Originally Posted by iMan_07

i understand that it is easier.....but we never covered half-angle and we're apparently not allowed to use it...double angle on the other hand, we used.....but i cant find a way to apply it to this question.

So what if you didn't cover it? Be ahead of your class and derive the half-angle identities. And you can do it using your double angle identities. Attempt it for abit, and if you reach to it or hit a dead-end, you can go here:
http://math.ucsd.edu/~wgarner/math4c.../halfangle.htm

Using the double angle identity for tan to find tan(x/2) is just complicated and prone to error.

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