What is the general solution for 3/(cos^2(x))=(7+4)/(cot(x)) ?

The general solution for 3/(cos^2(x))=(7+4)/(cot(x)) is x=(0.57693…)/2+pin,x= pi/2-(0.57693…)/2+pin

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3/(cos^2(x))=(7+4)/(cot(x))

\frac{3}{cos ^{2} ( x )} = \frac{7 + 4}{cot ( x )}

x = \frac{0.57693 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.57693 \dots}{2} + πn

Solution steps

\frac{3}{cos ^{2} ( x )} = \frac{7 + 4}{cot ( x )}

Subtract

\frac{7 + 4}{cot ( x )}

from both sides

\frac{3}{cos ^{2} ( x )} - \frac{11}{cot ( x )} = 0

Simplify

\frac{3}{cos ^{2} ( x )} - \frac{11}{cot ( x )} : \frac{3 cot ( x ) - 11 cos ^{2} ( x )}{cos ^{2} ( x ) cot ( x )}

\frac{3 cot ( x ) - 11 cos ^{2} ( x )}{cos ^{2} ( x ) cot ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 cot (x) - 11 cos^{2} (x) = 0

Express with sin, cos

\frac{3 cos ( x ) - 11 cos ^{2} ( x ) sin ( x )}{sin ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

3 cos (x) - 11 cos^{2} (x) sin (x) = 0

Factor

3 cos (x) - 11 cos^{2} (x) sin (x) : cos (x) (3 - 11 sin (x) cos (x))

cos (x) (3 - 11 sin (x) cos (x)) = 0

Solving each part separately

cos (x) = 0 or 3 - 11 sin (x) cos (x) = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

3 - 11 sin (x) cos (x) = 0 : x = \frac{arcsin ( \frac{6}{11} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{arcsin ( \frac{6}{11} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

x = \frac{arcsin ( \frac{6}{11} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{6}{11} )}{2} + πn

Show solutions in decimal form

x = \frac{0.57693 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.57693 \dots}{2} + πn

sin (x) = 0.01

sin (x) - sin (2 x) = sin^{3} (x)

- sin^{2} (x) = cos (2 x)

sin (x) = \frac{4}{5}, \frac{π}{2} \leq 0 < π, sin (2 x)

2 cos^{2} (θ) - 3 cos (θ) + 1 = 0, \frac{3 π}{2} < θ < 2 π

What is the general solution for 3/(cos^2(x))=(7+4)/(cot(x)) ?
The general solution for 3/(cos^2(x))=(7+4)/(cot(x)) is x=(0.57693…)/2+pin,x= pi/2-(0.57693…)/2+pin