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

The general solution for cos(x)tan(x)=5tan(x) is x=pin

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cos(x)tan(x)=5tan(x)

cos (x) tan (x) = 5 tan (x)

x = πn

Degrees

x = 0^{\circ} + 18 0^{\circ} n

Solution steps

cos (x) tan (x) = 5 tan (x)

Subtract

5 tan (x)

from both sides

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

Factor

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

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

Factor out common term

tan (x)

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

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

Solving each part separately

tan (x) = 0 or 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

cos (x) - 5 = 0 :

No Solution

cos (x) - 5 = 0

Move

5

to the right side

cos (x) - 5 = 0

Add

5

to both sides

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

Simplify

cos (x) = 5

cos (x) = 5

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = πn

- 5 sin (5 x) = 0

16 cos^{2} (θ) = 12

cos (2 x) = 2 cos (x)

sin (θ) + 3 cos (θ) = 1

6 tan^{2} (x) - 2 = 4

What is the general solution for cos(x)tan(x)=5tan(x) ?
The general solution for cos(x)tan(x)=5tan(x) is x=pin