What is the general solution for 3=7cos(pi/3 t) ?

The general solution for 3=7cos(pi/3 t) is t=(3*1.12788…)/pi+6n,t=6-(3*1.12788…)/pi+6n

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3=7cos(pi/3 t)

Solution

3 = 7 cos (\frac{π}{3} t)

Solution

t = \frac{3 \cdot 1.12788 \dots}{π} + 6 n, t = 6 - \frac{3 \cdot 1.12788 \dots}{π} + 6 n

Degrees

t = 61.71048 \dots^{\circ} + 343.77467 \dots^{\circ} n, t = 282.06419 \dots^{\circ} + 343.77467 \dots^{\circ} n

Solution steps

3 = 7 cos (\frac{π}{3} t)

Switch sides

7 cos (\frac{π}{3} t) = 3

Divide both sides by

7

7 cos (\frac{π}{3} t) = 3

Divide both sides by

7

\frac{7 cos ( \frac{π}{3} t )}{7} = \frac{3}{7}

Simplify

cos (\frac{π}{3} t) = \frac{3}{7}

cos (\frac{π}{3} t) = \frac{3}{7}

Apply trig inverse properties

cos (\frac{π}{3} t) = \frac{3}{7}

General solutions for

cos (\frac{π}{3} t) = \frac{3}{7}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

\frac{π}{3} t = arccos (\frac{3}{7}) + 2 πn, \frac{π}{3} t = 2 π - arccos (\frac{3}{7}) + 2 πn

\frac{π}{3} t = arccos (\frac{3}{7}) + 2 πn, \frac{π}{3} t = 2 π - arccos (\frac{3}{7}) + 2 πn

Solve

\frac{π}{3} t = arccos (\frac{3}{7}) + 2 πn : t = \frac{3 arccos ( \frac{3}{7} )}{π} + 6 n

\frac{π}{3} t = arccos (\frac{3}{7}) + 2 πn

Multiply both sides by

3

\frac{π}{3} t = arccos (\frac{3}{7}) + 2 πn

Multiply both sides by

3

3 \cdot \frac{π}{3} t = 3 arccos (\frac{3}{7}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{π}{3} t = 3 arccos (\frac{3}{7}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{π}{3} t : π t

3 \cdot \frac{π}{3} t

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 π}{3} t

Cancel the common factor:

3

= t π

Simplify

3 arccos (\frac{3}{7}) + 3 \cdot 2 πn : 3 arccos (\frac{3}{7}) + 6 πn

3 arccos (\frac{3}{7}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 3 arccos (\frac{3}{7}) + 6 πn

π t = 3 arccos (\frac{3}{7}) + 6 πn

π t = 3 arccos (\frac{3}{7}) + 6 πn

π t = 3 arccos (\frac{3}{7}) + 6 πn

Divide both sides by

π

π t = 3 arccos (\frac{3}{7}) + 6 πn

Divide both sides by

π

\frac{π t}{π} = \frac{3 arccos ( \frac{3}{7} )}{π} + \frac{6 πn}{π}

Simplify

t = \frac{3 arccos ( \frac{3}{7} )}{π} + 6 n

t = \frac{3 arccos ( \frac{3}{7} )}{π} + 6 n

Solve

\frac{π}{3} t = 2 π - arccos (\frac{3}{7}) + 2 πn : t = 6 - \frac{3 arccos ( \frac{3}{7} )}{π} + 6 n

\frac{π}{3} t = 2 π - arccos (\frac{3}{7}) + 2 πn

Multiply both sides by

3

\frac{π}{3} t = 2 π - arccos (\frac{3}{7}) + 2 πn

Multiply both sides by

3

3 \cdot \frac{π}{3} t = 3 \cdot 2 π - 3 arccos (\frac{3}{7}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{π}{3} t = 3 \cdot 2 π - 3 arccos (\frac{3}{7}) + 3 \cdot 2 πn

Simplify

3 \cdot \frac{π}{3} t : π t

3 \cdot \frac{π}{3} t

Multiply fractions:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 π}{3} t

Cancel the common factor:

3

= t π

Simplify

3 \cdot 2 π - 3 arccos (\frac{3}{7}) + 3 \cdot 2 πn : 6 π - 3 arccos (\frac{3}{7}) + 6 πn

3 \cdot 2 π - 3 arccos (\frac{3}{7}) + 3 \cdot 2 πn

Multiply the numbers:

3 \cdot 2 = 6

= 6 π - 3 arccos (\frac{3}{7}) + 6 πn

π t = 6 π - 3 arccos (\frac{3}{7}) + 6 πn

π t = 6 π - 3 arccos (\frac{3}{7}) + 6 πn

π t = 6 π - 3 arccos (\frac{3}{7}) + 6 πn

Divide both sides by

π

π t = 6 π - 3 arccos (\frac{3}{7}) + 6 πn

Divide both sides by

π

\frac{π t}{π} = \frac{6 π}{π} - \frac{3 arccos ( \frac{3}{7} )}{π} + \frac{6 πn}{π}

Simplify

\frac{π t}{π} = \frac{6 π}{π} - \frac{3 arccos ( \frac{3}{7} )}{π} + \frac{6 πn}{π}

Simplify

\frac{π t}{π} : t

\frac{π t}{π}

Cancel the common factor:

π

= t

Simplify

\frac{6 π}{π} - \frac{3 arccos ( \frac{3}{7} )}{π} + \frac{6 πn}{π} : 6 - \frac{3 arccos ( \frac{3}{7} )}{π} + 6 n

\frac{6 π}{π} - \frac{3 arccos ( \frac{3}{7} )}{π} + \frac{6 πn}{π}

Cancel

\frac{6 π}{π} : 6

\frac{6 π}{π}

Cancel the common factor:

π

= 6

= 6 - \frac{3 arccos ( \frac{3}{7} )}{π} + \frac{6 πn}{π}

Cancel

\frac{6 πn}{π} : 6 n

\frac{6 πn}{π}

Cancel the common factor:

π

= 6 n

= 6 - \frac{3 arccos ( \frac{3}{7} )}{π} + 6 n

t = 6 - \frac{3 arccos ( \frac{3}{7} )}{π} + 6 n

t = 6 - \frac{3 arccos ( \frac{3}{7} )}{π} + 6 n

t = 6 - \frac{3 arccos ( \frac{3}{7} )}{π} + 6 n

t = \frac{3 arccos ( \frac{3}{7} )}{π} + 6 n, t = 6 - \frac{3 arccos ( \frac{3}{7} )}{π} + 6 n

Show solutions in decimal form

t = \frac{3 \cdot 1.12788 \dots}{π} + 6 n, t = 6 - \frac{3 \cdot 1.12788 \dots}{π} + 6 n

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Frequently Asked Questions (FAQ)

What is the general solution for 3=7cos(pi/3 t) ?
The general solution for 3=7cos(pi/3 t) is t=(3*1.12788…)/pi+6n,t=6-(3*1.12788…)/pi+6n