4cos(3x)-1=2sin(3x)tan(3x)

Solution

4 cos (3 x) - 1 = 2 sin (3 x) tan (3 x)

x = \frac{0.84106 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.84106 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

Solution steps

4 cos (3 x) - 1 = 2 sin (3 x) tan (3 x)

Subtract

2 sin (3 x) tan (3 x)

from both sides

4 cos (3 x) - 1 - 2 sin (3 x) tan (3 x) = 0

Rewrite using trig identities

- 1 + \frac{- 2 + 6 cos ^{2} ( 3 x )}{cos ( 3 x )} = 0

Solve by substitution

cos (3 x) = \frac{2}{3}, cos (3 x) = - \frac{1}{2}

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

cos (3 x) = - \frac{1}{2} : x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

Combine all the solutions

x = \frac{arccos ( \frac{2}{3} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{arccos ( \frac{2}{3} )}{3} + \frac{2 πn}{3}, x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

Show solutions in decimal form

x = \frac{0.84106 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{3} - \frac{0.84106 \dots}{3} + \frac{2 πn}{3}, x = \frac{2 π}{9} + \frac{2 πn}{3}, x = \frac{4 π}{9} + \frac{2 πn}{3}

tan (θ) = \frac{15}{20}

- cos (x) = 0.9

3 cos (θ) - 2 sin (θ) cos (θ) = 0

sec (x) + sin^{2} (x) + cos^{2} (x) = tan^{2} (x)

4 sin^{2} (x) - 3 = 4, sin (2 x) - 3 = 0