How do you use the half angle identity to find exact value of tan 112.5?

1 Answer
Nghi N.
Nov 9, 2015

Find exact value of #tan (112.5)#

Explanation:

Call tan (112.5) = tan x.
tan 2x = tan (225) = tan (45 + 180) = tan 45 = 1
Apply the trig identity: #tan 2x = (2tan x)/(1 - tan^2 x)#, we get:
#1 = (2tan x)/(1 - tan^2 x)#
#1 - tan^2 x = 2tan x#.
Solve the quadratic equation:
#tan^2 x + 2tan x - 1 = 0#
#D = d^2 = b^2 - 4ac = 4 + 4 = 8# --> #d = +- 2sqrt2#
Ther3 are 2 real roots:
#tan x = -2/2 +- 2sqrt2/2 = -1 +- sqrt2.#
Since (tan 112.5) is negative (Quadrant II), then
#tan x = -1 - sqrt2#
Check by calculator.
sin (112.5) = -2.14
#(-1 - sqrt2) = -1 - 1,41 = - 2.41#. OK

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