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Popular Trigonometry >

solvefor x,T(6)=3.15cos(pi/6 x)+19.15

Solution

solvefor

Solution

x = \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + 12 n, x = - \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + 12 n

Solution steps

T (6) = 3.15 cos (\frac{π}{6} x) + 19.15

Switch sides

3.15 cos (\frac{π}{6} x) + 19.15 = T \cdot 6

Multiply both sides by

100

3.15 cos (\frac{π}{6} x) + 19.15 = T \cdot 6

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

3.15 cos (\frac{π}{6} x) \cdot 100 + 19.15 \cdot 100 = T \cdot 6 \cdot 100

Refine

315 cos (\frac{π}{6} x) + 1915 = 600 T

315 cos (\frac{π}{6} x) + 1915 = 600 T

Move

1915

to the right side

315 cos (\frac{π}{6} x) + 1915 = 600 T

Subtract

1915

from both sides

315 cos (\frac{π}{6} x) + 1915 - 1915 = 600 T - 1915

Simplify

315 cos (\frac{π}{6} x) = 600 T - 1915

315 cos (\frac{π}{6} x) = 600 T - 1915

Divide both sides by

315

315 cos (\frac{π}{6} x) = 600 T - 1915

Divide both sides by

315

\frac{315 cos ( \frac{π}{6} x )}{315} = \frac{600 T}{315} - \frac{1915}{315}

Simplify

\frac{315 cos ( \frac{π}{6} x )}{315} = \frac{600 T}{315} - \frac{1915}{315}

Simplify

\frac{315 cos ( \frac{π}{6} x )}{315} : cos (\frac{π}{6} x)

\frac{315 cos ( \frac{π}{6} x )}{315}

Divide the numbers:

\frac{315}{315} = 1

= cos (\frac{π}{6} x)

Simplify

\frac{600 T}{315} - \frac{1915}{315} : \frac{120 T - 383}{63}

\frac{600 T}{315} - \frac{1915}{315}

Apply rule

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

= \frac{600 T - 1915}{315}

Factor

600 T - 1915 : 5 (120 T - 383)

600 T - 1915

Rewrite as

= 5 \cdot 120 T - 5 \cdot 383

Factor out common term

5

= 5 (120 T - 383)

= \frac{5 ( 120 T - 383 )}{315}

Cancel the common factor:

5

= \frac{120 T - 383}{63}

cos (\frac{π}{6} x) = \frac{120 T - 383}{63}

cos (\frac{π}{6} x) = \frac{120 T - 383}{63}

cos (\frac{π}{6} x) = \frac{120 T - 383}{63}

Apply trig inverse properties

cos (\frac{π}{6} x) = \frac{120 T - 383}{63}

General solutions for

cos (\frac{π}{6} x) = \frac{120 T - 383}{63}

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

\frac{π}{6} x = arccos (\frac{120 T - 383}{63}) + 2 πn, \frac{π}{6} x = - arccos (\frac{120 T - 383}{63}) + 2 πn

\frac{π}{6} x = arccos (\frac{120 T - 383}{63}) + 2 πn, \frac{π}{6} x = - arccos (\frac{120 T - 383}{63}) + 2 πn

Solve

\frac{π}{6} x = arccos (\frac{120 T - 383}{63}) + 2 πn : x = \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + 12 n

\frac{π}{6} x = arccos (\frac{120 T - 383}{63}) + 2 πn

Multiply both sides by

6

\frac{π}{6} x = arccos (\frac{120 T - 383}{63}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x = 6 arccos (\frac{120 T - 383}{63}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x = 6 arccos (\frac{120 T - 383}{63}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiply fractions:

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

= \frac{6 π}{6} x

Cancel the common factor:

6

= x π

Simplify

6 arccos (\frac{120 T - 383}{63}) + 6 \cdot 2 πn : 6 arccos (\frac{120 T - 383}{63}) + 12 πn

6 arccos (\frac{120 T - 383}{63}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= 6 arccos (\frac{120 T - 383}{63}) + 12 πn

π x = 6 arccos (\frac{120 T - 383}{63}) + 12 πn

π x = 6 arccos (\frac{120 T - 383}{63}) + 12 πn

π x = 6 arccos (\frac{120 T - 383}{63}) + 12 πn

Divide both sides by

π

π x = 6 arccos (\frac{120 T - 383}{63}) + 12 πn

Divide both sides by

π

\frac{π x}{π} = \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + \frac{12 πn}{π}

Simplify

x = \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + 12 n

x = \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + 12 n

Solve

\frac{π}{6} x = - arccos (\frac{120 T - 383}{63}) + 2 πn : x = - \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + 12 n

\frac{π}{6} x = - arccos (\frac{120 T - 383}{63}) + 2 πn

Multiply both sides by

6

\frac{π}{6} x = - arccos (\frac{120 T - 383}{63}) + 2 πn

Multiply both sides by

6

6 \cdot \frac{π}{6} x = - 6 arccos (\frac{120 T - 383}{63}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x = - 6 arccos (\frac{120 T - 383}{63}) + 6 \cdot 2 πn

Simplify

6 \cdot \frac{π}{6} x : π x

6 \cdot \frac{π}{6} x

Multiply fractions:

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

= \frac{6 π}{6} x

Cancel the common factor:

6

= x π

Simplify

- 6 arccos (\frac{120 T - 383}{63}) + 6 \cdot 2 πn : - 6 arccos (\frac{120 T - 383}{63}) + 12 πn

- 6 arccos (\frac{120 T - 383}{63}) + 6 \cdot 2 πn

Multiply the numbers:

6 \cdot 2 = 12

= - 6 arccos (\frac{120 T - 383}{63}) + 12 πn

π x = - 6 arccos (\frac{120 T - 383}{63}) + 12 πn

π x = - 6 arccos (\frac{120 T - 383}{63}) + 12 πn

π x = - 6 arccos (\frac{120 T - 383}{63}) + 12 πn

Divide both sides by

π

π x = - 6 arccos (\frac{120 T - 383}{63}) + 12 πn

Divide both sides by

π

\frac{π x}{π} = - \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + \frac{12 πn}{π}

Simplify

x = - \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + 12 n

x = - \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + 12 n

x = \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + 12 n, x = - \frac{6 arccos ( \frac{120 T - 383}{63} )}{π} + 12 n

Graph

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