Question

Can someone please help? This is the last question on my worksheet and I'm so stuck and completely brain dead. :(
Find the exact value by using a half-angle identity.

tan(7pi/8)

Answer 1

hint:

use the identity shown on this page

http://www.mathwords.com/h/half_angle_identities.htm

and notice how (1/2)*(7pi/4) = 7pi/8

so that means you'll be locating 7pi/4 on the unit circle

Answer 2

Ah thank you for even looking at this! I've been here for like an hour hoping to get help...

Answer 3

Sorry for the wait. Does the hint and/or the page help at all?

Answer 4

So it's sqrt2/2?

Answer 5

the unit circle is this

http://www.regentsprep.org/regents/math/algtrig/ATT5/600px-Unit_circle_angles_svg.jpg

Answer 6

Right and then I see that the one for 7pi/4 is sqrt2/2 and -sqrt2/2

Answer 7

notice how tan(7pi/8) = -0.41421 when we use a calculator and make sure it's in radian mode

so that means that the final answer must be negative

Answer 8

that means if we use the identity

\[\large \tan\left(\frac{x}{2}\right) = \pm \sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\]

then we have to use the negative form of it and we know to use this version below

\[\large \tan\left(\frac{x}{2}\right) = -\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\]


making sense so far?

Answer 9

sure but then what was the point of the unit circle?

Answer 10

what were we looking for if we're using this formula to solve the equation?

Answer 11

well we use the unit circle to determine cos(x)

in this case, x = 7pi/4

so you have the right idea to get cos(7pi/4) = sqrt(2)/2

Answer 12

So now we plug sqrt2/2 in then...ok that makes sense

Answer 13

so we'll have

\[\large \tan\left(\frac{x}{2}\right) = -\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\]

\[\large \tan\left(\frac{1}{2}*x\right) = -\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}\]

\[\large \tan\left(\frac{1}{2}*\frac{7\pi}{4}\right) = -\sqrt{\frac{1-\cos\left(\frac{7\pi}{4}\right)}{1+\cos\left(\frac{7\pi}{4}\right)}}\]

\[\large \tan\left(\frac{1}{2}*\frac{7\pi}{4}\right) = -\sqrt{\frac{1-\frac{\sqrt{2}}{2}}{1+\frac{\sqrt{2}}{2}}}\]

see how I'm getting all this so far?