How do you use a half-angle formula to simplify $tan [\frac{9 π}{8}]$ $tan[ (9pi) / 8 ]$ ?

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How do you use a half-angle formula to simplify $tan [\frac{9 π}{8}]$ $tan[ (9pi) / 8 ]$ ?

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Answer 1 · 2015-08-02T21:47:37

Simplify $tan (\frac{9 π}{8})$ $tan ((9pi)/8)$

Explanation:

$tan (\frac{9 π}{8}) = tan (\frac{π}{8} + π)$ $tan ((9pi)/8) = tan (pi/8 + pi)$ = tan pi/8
Call tan $\frac{π}{8} = t$ $pi/8 = t$
$tan 2 t = tan (\frac{2 π}{8}) = tan (\frac{π}{4}) = 1$ $tan 2t = tan ((2pi)/8) = tan (pi/4) = 1$
Use trig identity: $tan 2 t = 2 \frac{t}{1 - t^{2}}$ $tan 2t = 2t/(1 - t^2)$
$tan (\frac{π}{4}) = 1 = \frac{2 t}{1 - t^{2}}$ $tan (pi/4) = 1 = (2t)/(1 - t^2)$ --> $1 - t^{2} = 2 t$ $1 - t^2 = 2t$ -->

$t^{2} + 2 t - 1 = 0$ $t^2 + 2t - 1 = 0$
$D = d^{2} = b^{2} - 4 a c = 4 + 4 = 8$ $D = d^2 = b^2 - 4ac = 4 + 4 = 8$ --> $d = \pm 2 \sqrt{2}$ $d = +- 2sqrt2$

$tan (\frac{9 π}{8}) = tan (\frac{π}{8}) = t = - 1 \pm \sqrt{2}$ $tan ((9pi)/8) = tan (pi/8) = t = - 1 +- sqrt2$

Since the arc $\frac{9 π}{8}$ $(9pi)/8$ is in Quadrant III, then only the positive answer is accepted.
$\frac{tan (9 π)}{8} = - 1 + \sqrt{2}$ $tan (9pi)/8 = - 1 + sqrt2$ = 0.414

Check with calculator --> tan (22.5) = 0.414 OK

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How do you use a half-angle formula to simplify $tan [\frac{9 π}{8}]$ $tan[ (9pi) / 8 ]$ ?

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How do you use a half-angle formula to simplify $tan [\frac{9 π}{8}]$ $tan[ (9pi) / 8 ]$ ?