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

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

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tan(θ)cos(θ)+cos(θ)=0

tan (θ) cos (θ) + cos (θ) = 0

θ = \frac{3 π}{4} + πn

Degrees

θ = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

tan (θ) cos (θ) + cos (θ) = 0

Factor

tan (θ) cos (θ) + cos (θ) : cos (θ) (tan (θ) + 1)

tan (θ) cos (θ) + cos (θ)

Factor out common term

cos (θ)

= cos (θ) (tan (θ) + 1)

cos (θ) (tan (θ) + 1) = 0

Solving each part separately

cos (θ) = 0 or tan (θ) + 1 = 0

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

cos (θ) = 0

General solutions for

cos (θ) = 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}

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

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

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

tan (θ) + 1 = 0

Move

1

to the right side

tan (θ) + 1 = 0

Subtract

1

from both sides

tan (θ) + 1 - 1 = 0 - 1

Simplify

tan (θ) = - 1

tan (θ) = - 1

General solutions for

tan (θ) = - 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}

θ = \frac{3 π}{4} + πn

θ = \frac{3 π}{4} + πn

Combine all the solutions

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

Since the equation is undefined for:

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

θ = \frac{3 π}{4} + πn

sin (x) = cos (x), 0 \leq x \leq 2 π

3 cos (θ) = 0

2 sin^{2} (θ) + sin (θ) - 9 = 2 sin (θ) - 8

cos (θ) = \frac{5}{3}

- 16 sin (x) + 8 sin (2 x) = 0

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