What is the general solution for cos(1/2 x)+1=0 ?

The general solution for cos(1/2 x)+1=0 is x=2pi+4pin

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cos(1/2 x)+1=0

cos (\frac{1}{2} x) + 1 = 0

x = 2 π + 4 πn

Degrees

x = 36 0^{\circ} + 72 0^{\circ} n

Solution steps

cos (\frac{1}{2} x) + 1 = 0

Move

1

to the right side

cos (\frac{1}{2} x) + 1 = 0

Subtract

1

from both sides

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

Simplify

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

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

General solutions for

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

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}

\frac{1}{2} x = π + 2 πn

\frac{1}{2} x = π + 2 πn

Solve

\frac{1}{2} x = π + 2 πn : x = 2 π + 4 πn

\frac{1}{2} x = π + 2 πn

Multiply both sides by

2

\frac{1}{2} x = π + 2 πn

Multiply both sides by

2

2 \cdot \frac{1}{2} x = 2 π + 2 \cdot 2 πn

Simplify

x = 2 π + 4 πn

x = 2 π + 4 πn

x = 2 π + 4 πn

cos (x) + 5 = 5

cos^{2} (x) + cos (x) + 1 = 0

cos (x) = cos (\frac{x}{2})

3 tan^{2} (x) - 3 tan (x) = 0

- 4 sec^{2} (x) + 19 = - 4 tan (x)

What is the general solution for cos(1/2 x)+1=0 ?
The general solution for cos(1/2 x)+1=0 is x=2pi+4pin