What is the general solution for 7cos(x)tan(x)-5tan(x)=0 ?

The general solution for 7cos(x)tan(x)-5tan(x)=0 is x=pin,x=0.77519…+2pin,x=2pi-0.77519…+2pin

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7cos(x)tan(x)-5tan(x)=0

7 cos (x) tan (x) - 5 tan (x) = 0

x = πn, x = 0.77519 \dots + 2 πn, x = 2 π - 0.77519 \dots + 2 πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n, x = 44.41530 \dots^{\circ} + 36 0^{\circ} n, x = 315.58469 \dots^{\circ} + 36 0^{\circ} n

Solution steps

7 cos (x) tan (x) - 5 tan (x) = 0

Factor

7 cos (x) tan (x) - 5 tan (x) : tan (x) (7 cos (x) - 5)

7 cos (x) tan (x) - 5 tan (x)

Factor out common term

tan (x)

= tan (x) (7 cos (x) - 5)

tan (x) (7 cos (x) - 5) = 0

Solving each part separately

tan (x) = 0 or 7 cos (x) - 5 = 0

tan (x) = 0 : x = πn

tan (x) = 0

General solutions for

tan (x) = 0

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Solve

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

7 cos (x) - 5 = 0 : x = arccos (\frac{5}{7}) + 2 πn, x = 2 π - arccos (\frac{5}{7}) + 2 πn

7 cos (x) - 5 = 0

Move

5

to the right side

7 cos (x) - 5 = 0

Add

5

to both sides

7 cos (x) - 5 + 5 = 0 + 5

Simplify

7 cos (x) = 5

7 cos (x) = 5

Divide both sides by

7

7 cos (x) = 5

Divide both sides by

7

\frac{7 cos ( x )}{7} = \frac{5}{7}

Simplify

cos (x) = \frac{5}{7}

cos (x) = \frac{5}{7}

Apply trig inverse properties

cos (x) = \frac{5}{7}

General solutions for

cos (x) = \frac{5}{7}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{5}{7}) + 2 πn, x = 2 π - arccos (\frac{5}{7}) + 2 πn

x = arccos (\frac{5}{7}) + 2 πn, x = 2 π - arccos (\frac{5}{7}) + 2 πn

Combine all the solutions

x = πn, x = arccos (\frac{5}{7}) + 2 πn, x = 2 π - arccos (\frac{5}{7}) + 2 πn

Show solutions in decimal form

x = πn, x = 0.77519 \dots + 2 πn, x = 2 π - 0.77519 \dots + 2 πn

sec^{2} (x) - 6 sec (x) = 0

9 sec^{2} (x) tan (x) = 12 tan (x)

cos (x) = - \frac{4}{7}

(8 cos (θ) - 4) (5 sin (θ) - 5) = 0

arctan (x) = \frac{π}{3}

What is the general solution for 7cos(x)tan(x)-5tan(x)=0 ?
The general solution for 7cos(x)tan(x)-5tan(x)=0 is x=pin,x=0.77519…+2pin,x=2pi-0.77519…+2pin