What is the general solution for tan(x)cos(x)+cos(x)=0 ?

The general solution for tan(x)cos(x)+cos(x)=0 is x=(3pi)/4+pin

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tan(x)cos(x)+cos(x)=0

tan (x) cos (x) + cos (x) = 0

x = \frac{3 π}{4} + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

tan (x) cos (x) + cos (x) = 0

Factor

tan (x) cos (x) + cos (x) : cos (x) (tan (x) + 1)

tan (x) cos (x) + cos (x)

Factor out common term

cos (x)

= cos (x) (tan (x) + 1)

cos (x) (tan (x) + 1) = 0

Solving each part separately

cos (x) = 0 or tan (x) + 1 = 0

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

cos (x) = 0

General solutions for

cos (x) = 0

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

tan (x) + 1 = 0 : x = \frac{3 π}{4} + πn

tan (x) + 1 = 0

Move

1

to the right side

tan (x) + 1 = 0

Subtract

1

from both sides

tan (x) + 1 - 1 = 0 - 1

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 1

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 = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Combine all the solutions

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

Since the equation is undefined for:

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

x = \frac{3 π}{4} + πn

sin^{2} (x) (tan^{2} (x)) = 1 - sin^{2} (x)

2 sin^{2} (x) = 3 - 5 cos (x)

- 3 cos (2 θ) - 18 sin (θ) - 8 = - 5 sin (θ) - 6

cos (x) (2 cos (x) + 1) = 0

tan^{2} (x) + 4 tan (x) = 3

What is the general solution for tan(x)cos(x)+cos(x)=0 ?
The general solution for tan(x)cos(x)+cos(x)=0 is x=(3pi)/4+pin