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Hey guys! Does anybody know the answer to tan2x-2cosx=0

User Narayanan
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1 Answer

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If the equation is tan(2x) - 2 cos(x) = 0:

We have

tan(2x) = sin(2x) / cos(2x)

sin(2x) = 2 sin(x) cos(x)

So, rewriting the equation gives

tan(2x) - 2 cos(x) = 0

2 sin(x) cos(x) / cos(2x) - 2 cos(x) cos(2x) / cos(2x)= 0

2 cos(x) • (sin(x) - cos(2x)) / cos(2x) = 0

The left side is 0 whenever the numerator is 0:

2 cos(x) = 0 or sin(x) - cos(2x) = 0

cos(x) = 0 or sin(x) - (1 - 2 sin²(x)) = 0

cos(x) = 0 or 2 sin²(x) + sin(x) - 1 = 0

cos(x) = 0 or (2 sin(x) - 1) (sin(x) + 1) = 0

cos(x) = 0 or 2 sin(x) - 1 = 0 or sin(x) + 1 = 0

cos(x) = 0 or sin(x) = 1/2 or sin(x) = -1

Solving each case gives 3 families of solutions:

[x = π/2 + 2 or x = -π/2 + 2] or

[x = π/6 + 2 or x = 5π/6 + 2] or

[x = -π/2 + 2]

where n is any integer. The last family is already accounted for in the first, so

x = π/2 + 2 or x = -π/2 + 2 or x = π/6 + 2 or x = 5π/6 + 2

User Jtabuloc
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8.2k points

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