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Popular >

tan(x)=(5+cos(x))/(6sin(x)cos(x))

Solution

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

Solution

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

+1

Solution steps

tan (x) = \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

Subtract

\frac{5 + cos ( x )}{6 sin ( x ) cos ( x )}

from both sides

tan (x) - \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )} = 0

Simplify

tan (x) - \frac{5 + cos ( x )}{6 sin ( x ) cos ( x )} : \frac{6 tan ( x ) sin ( x ) cos ( x ) - 5 - cos ( x )}{6 sin ( x ) cos ( x )}

\frac{6 tan ( x ) sin ( x ) cos ( x ) - 5 - cos ( x )}{6 sin ( x ) cos ( x )} = 0

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

6 tan (x) sin (x) cos (x) - 5 - cos (x) = 0

Express with sin, cos

6 \cdot \frac{sin ( x )}{cos ( x )} sin (x) cos (x) - 5 - cos (x) = 0

Simplify

6 \cdot \frac{sin ( x )}{cos ( x )} sin (x) cos (x) - 5 - cos (x) : 6 sin^{2} (x) - 5 - cos (x)

6 sin^{2} (x) - 5 - cos (x) = 0

Add

cos (x)

to both sides

6 sin^{2} (x) - 5 = cos (x)

Square both sides

(6 sin^{2} (x) - 5)^{2} = cos^{2} (x)

Subtract

cos^{2} (x)

from both sides

(6 sin^{2} (x) - 5)^{2} - cos^{2} (x) = 0

Factor

(6 sin^{2} (x) - 5)^{2} - cos^{2} (x) : (6 sin^{2} (x) - 5 + cos (x)) (6 sin^{2} (x) - 5 - cos (x))

(6 sin^{2} (x) - 5 + cos (x)) (6 sin^{2} (x) - 5 - cos (x)) = 0

Solving each part separately

6 sin^{2} (x) - 5 + cos (x) = 0 or 6 sin^{2} (x) - 5 - cos (x) = 0

6 sin^{2} (x) - 5 + cos (x) = 0 : x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

6 sin^{2} (x) - 5 - cos (x) = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

Combine all the solutions

x = arccos (- \frac{1}{3}) + 2 πn, x = - arccos (- \frac{1}{3}) + 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

Verify solutions by plugging them into the original equation

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = arccos (\frac{1}{3}) + 2 πn, x = 2 π - arccos (\frac{1}{3}) + 2 πn

Show solutions in decimal form

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn, x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

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