Given tanx=8/15 when pi < x < 3pi/2, find tan x/2.
Use tan (2A) = 2tanA/(1 - tan^2 A)
or tanx = 2tan (x/2)(1 - tan^2 (x/2)) , let tan x/2 = y for easier typing
8/15 = 2y/(1 - y^2)
30y = 8 - 8y^2
8y^2 + 30y - 8 = 0
4y^2 + 15y - 4 = 0
(4y - 1)(y + 4) = 0
y = 1/4 or y = -4
tan x/2 = 1/4 or tan x/2 = -4
but x was in quadrant III , so x/2 must be in quadrant II
and in II, the tangent is negative,
so tan x/2 = -4
if tanx = 8/15 in QIII then
sinx = -8/√161
cosx = -15/√161
so
tanx/2 = (1-cosx)/sinx = (1 + 8/√161)/(-15/√161) = -(8+√161)/15
makes sense, since x/2 will be in QII where tangent is negative
oops - go with Reiny.
Do you see my mistake?
Tanx = -8/-15 = 8/15.
X = 28.1o S. of W. (Q3). = 208.1o CCW.
x/2 = 208.1/2 = 104o CCW = 76o N. of W.
Tan(x/2) = Tan104 = -4.0.
To find tan(x/2), we can use the Half-Angle Identity for tangent. The formula for tan(x/2) is:
tan(x/2) = ±√((1 - cos(x))/(1 + cos(x)))
First, let's find the value of cos(x). We know that tan(x) = 8/15 in the given interval, and tan(x) is positive, which implies that x is in the second quadrant. In the second quadrant, the value of tan(x) is positive and sin(x) is positive, but cos(x) is negative.
To find cos(x), we can use the Pythagorean Identity for tangent:
tan(x) = sin(x)/cos(x)
Given that tan(x) = 8/15, we can substitute the values into the identity and solve for cos(x):
8/15 = sin(x)/cos(x)
Cross-multiplying:
8*cos(x) = 15*sin(x)
Using the Pythagorean Identity:
8*cos(x) = 15*√(1 - cos^2(x))
Squaring both sides:
64*cos^2(x) = 225*(1 - cos^2(x))
Expanding:
64*cos^2(x) = 225 - 225*cos^2(x)
Combining like terms:
289*cos^2(x) = 225
Dividing both sides by 289:
cos^2(x) = 225/289
Taking the square root of both sides:
cos(x) = ±√(225/289)
Since cos(x) is negative in the second quadrant, we take the negative value:
cos(x) = -√(225/289) = -15/17
Now that we have the value of cos(x), we can substitute it into the formula for tan(x/2):
tan(x/2) = ±√((1 - cos(x))/(1 + cos(x)))
tan(x/2) = ±√((1 - (-15/17))/(1 + (-15/17)))
Simplifying:
tan(x/2) = ±√((17 + 15)/(17 - 15))
tan(x/2) = ±√(32/2)
tan(x/2) = ±√16
Finally, the value of tan(x/2) is either +4 or -4, depending on the sign.