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atkelly
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#1
Report Thread starter 14 years ago
#1
Hi,

Can anyone help me with this. Solve this equation:

cot(x) - cosec^2(x) + 3 = 0

Thanks.

AK
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misshn
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#2
Report 14 years ago
#2
cosx/sinx - 1/(sinx)^2 +3 = 0
=> cosx sinx - 1 + 3(sinx)^2 = 0 (sinx <> 0)
=> 3(sinx)^2 + cosx sinx - (cosx)^2 - (sinx)^2 = 0
=> 2(sinx)^2 + cosx sinx - (cosx)^2 = 0
=> (2sinx - cosx)(sinx + cosx) = 0
=> cosx = 2sinx or cosx = -sinx
=> tanx = 1/2 or tanx = -1
=> x = arctan(1/2) + kpi or x = -pi/4 + kpi
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Syncman
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#3
Report 14 years ago
#3
(Original post by atkelly)
Hi,

Can anyone help me with this. Solve this equation:

cot(x) - cosec^2(x) + 3 = 0

Thanks.

AK
cosec^2 x = cot^2 x + 1

So.
cot x - (cot^2 x + 1) + 3 = 0
quadratic in cot x
cot^2 x - cot x - 2 = 0 (took all to other side so cot^2 x is positive)
Ley m = cot x
m^2 - m - 2 = 0
(m - 2)(m + 1)
So either cot x = 2 or cot x = -1
tan x = 1/2 or tanx = -1

the find it in degree's or radians.
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Nima
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#4
Report 14 years ago
#4
(Original post by atkelly)
cot(x) - cosec^2(x) + 3 = 0
cot(x) - cosec^2x + 3 = 0
---> cot(x) - [1 + cot^2(x)] + 3 = 0
---> -cot^2(x) + cot(x) + 2 = 0
---> cot^2(x) - cot(x) - 2 = 0
---> [cot(x) + 1][cot(x) - 2] = 0
---> cot(x) = -1, cot(x) = 2
---> tan(x) = -1, tan(x) = 1/2

Finding solutions from tan(x) = -1:

x = tan^-1(-1) = -Pi/4 Rads
x = -Pi/4 + kPi Rads

Where k is an integer.

Finding solutions from tan(x) = 1/2:

x = tan^-1(1/2) = 0.4816 Rads
x = 0.4816 + kPi Rads

Where k is an integer.
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