Exact value of angle

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March 20th 2006, 10:09 AM
macca101

Exact value of angle

$ \mbox{Given that} \ sec\theta = 2 , tan\theta = -\sqrt{3} \ and \ -\pi <\theta< \pi $
Find the exact value of the angle $ \theta $ in radians and justify your answer.

I'm stuck on this one why can't I just use the arctan function on my calculator? I know I can't that would be too simple. Can any one give me a clue ?
March 20th 2006, 03:05 PM
topsquark

Quote:

Originally Posted by macca101

$ \mbox{Given that} \ sec\theta = 2 , tan\theta = -\sqrt{3} \ and \ -\pi <\theta< \pi $
Find the exact value of the angle $ \theta $ in radians and justify your answer.

I'm stuck on this one why can't I just use the arctan function on my calculator? I know I can't that would be too simple. Can any one give me a clue ?

Well, cos = 1/sec so we have that $cos\theta= 1/2$ , which leads us to $tan\theta= \frac{sin\theta}{cos\theta}=-\sqrt3$ which implies that $sin\theta=-\sqrt3/2$ . Where in $-\pi <\theta< \pi$ do we find such an angle? $\theta=- \pi/3$ is the only one I can think of.

The reason that just taking the inverse tangent won't work is that you will get a list of possibles and the answer is buried in there. However, you could do the inverse tangent and inverse secant and compare the two lists...(rather like what I did with the sine and cosine values.)

-Dan
March 20th 2006, 03:06 PM
ThePerfectHacker

Quote:

Originally Posted by macca101

$ \mbox{Given that} \ sec\theta = 2 , tan\theta = -\sqrt{3} \ and \ -\pi <\theta< \pi $
Find the exact value of the angle $ \theta $ in radians and justify your answer.

I'm stuck on this one why can't I just use the arctan function on my calculator? I know I can't that would be too simple. Can any one give me a clue ?

There are some anlges that you need to have remorzied to be able to answer this question.
Notice that,
$\sec x=2$
Thus,
$\cos x=1/2$
That happens, when $x=\frac{\pi}{3}$ -memorized.

Notice that, $\tan x=\sqrt{3}$
happens when $\frac{\pi}{3}$ -memorized.
But, problem ask for $\tan x=-\sqrt{3}$
Thus, since $\tan (-x)=-\tan x$
We have that $x=-\frac{\pi}{3}$
March 20th 2006, 03:50 PM
Rich B.

Hi:

In evaluating arctan [-sqrt(3)], your calculator will provide a decimal approximation to the tune of -1.0471975511 ± a few decimal places. But the problem is specific in calling for an EXACT value of theta which, as it happens, is irrational and therefore with infinitely many non-repeating digits to the right of the decimal point. And this is why you are well advised by PHckr, to commit exact trigonometric values of certain key angles to memory.

Regards,

Rich B
March 21st 2006, 01:43 AM
macca101

Thanks