What is the general solution for -1=cos(pit) ?

The general solution for -1=cos(pit) is t=1+2n

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-1=cos(pit)

- 1 = cos (π t)

t = 1 + 2 n

Degrees

t = 57.29577 \dots^{\circ} + 114.59155 \dots^{\circ} n

Solution steps

- 1 = cos (π t)

Switch sides

cos (π t) = - 1

General solutions for

cos (π t) = - 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}

π t = π + 2 πn

π t = π + 2 πn

Solve

π t = π + 2 πn : t = 1 + 2 n

π t = π + 2 πn

Divide both sides by

π

π t = π + 2 πn

Divide both sides by

π

\frac{π t}{π} = \frac{π}{π} + \frac{2 πn}{π}

Simplify

\frac{π t}{π} = \frac{π}{π} + \frac{2 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{π}{π} + \frac{2 πn}{π} : 1 + 2 n

\frac{π}{π} + \frac{2 πn}{π}

Apply rule

\frac{a}{a} = 1

\frac{π}{π} = 1

= 1 + \frac{2 πn}{π}

Cancel

\frac{2 πn}{π} : 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= 1 + 2 n

t = 1 + 2 n

t = 1 + 2 n

t = 1 + 2 n

t = 1 + 2 n

2 tan^{2} (θ) = 6

cos^{2} (θ + \frac{π}{6}) = \frac{1}{2}

2 sin (2 x) = 2 sin (x)

tan (x) = \frac{1}{7}

cos^{2} (x) = 0.1

What is the general solution for -1=cos(pit) ?
The general solution for -1=cos(pit) is t=1+2n