Trigonometry Examples

$t = 50 - 15 \cos (\frac{x π}{6})$

Step 1

Rewrite the equation as $50 - 15 \cos (\frac{x π}{6}) = t$ .

$50 - 15 \cos (\frac{x π}{6}) = t$

Step 2

Subtract $50$ from both sides of the equation.

$- 15 \cos (\frac{x π}{6}) = t - 50$

Step 3

Divide each term in $- 15 \cos (\frac{x π}{6}) = t - 50$ by $- 15$ and simplify.

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Step 3.1

Divide each term in $- 15 \cos (\frac{x π}{6}) = t - 50$ by $- 15$ .

$\frac{- 15 \cos (\frac{x π}{6})}{- 15} = \frac{t}{- 15} + \frac{- 50}{- 15}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 15$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{- 15 \cos (\frac{x π}{6})}{- 15} = \frac{t}{- 15} + \frac{- 50}{- 15}$

Step 3.2.1.2

Divide $\cos (\frac{x π}{6})$ by $1$ .

$\cos (\frac{x π}{6}) = \frac{t}{- 15} + \frac{- 50}{- 15}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative in front of the fraction.

$\cos (\frac{x π}{6}) = - \frac{t}{15} + \frac{- 50}{- 15}$

Step 3.3.1.2

Cancel the common factor of $- 50$ and $- 15$ .

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Step 3.3.1.2.1

Factor $- 5$ out of $- 50$ .

$\cos (\frac{x π}{6}) = - \frac{t}{15} + \frac{- 5 (10)}{- 15}$

Step 3.3.1.2.2

Cancel the common factors.

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Step 3.3.1.2.2.1

Factor $- 5$ out of $- 15$ .

$\cos (\frac{x π}{6}) = - \frac{t}{15} + \frac{- 5 \cdot 10}{- 5 \cdot 3}$

Step 3.3.1.2.2.2

Cancel the common factor.

$\cos (\frac{x π}{6}) = - \frac{t}{15} + \frac{- 5 \cdot 10}{- 5 \cdot 3}$

Step 3.3.1.2.2.3

Rewrite the expression.

$\cos (\frac{x π}{6}) = - \frac{t}{15} + \frac{10}{3}$

Step 4

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$\frac{x π}{6} = \arccos (- \frac{t}{15} + \frac{10}{3})$

Step 5

Multiply both sides of the equation by $\frac{6}{π}$ .

$\frac{6}{π} \cdot \frac{x π}{6} = \frac{6}{π} \arccos (- \frac{t}{15} + \frac{10}{3})$

Step 6

Simplify both sides of the equation.

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Step 6.1

Simplify the left side.

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Step 6.1.1

Simplify $\frac{6}{π} \cdot \frac{x π}{6}$ .

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Step 6.1.1.1

Cancel the common factor of $6$ .

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Step 6.1.1.1.1

Cancel the common factor.

$\frac{6}{π} \cdot \frac{x π}{6} = \frac{6}{π} \arccos (- \frac{t}{15} + \frac{10}{3})$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{π} (x π) = \frac{6}{π} \arccos (- \frac{t}{15} + \frac{10}{3})$

Step 6.1.1.2

Cancel the common factor of $π$ .

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Step 6.1.1.2.1

Factor $π$ out of $x π$ .

$\frac{1}{π} (π x) = \frac{6}{π} \arccos (- \frac{t}{15} + \frac{10}{3})$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{π} (π x) = \frac{6}{π} \arccos (- \frac{t}{15} + \frac{10}{3})$

Step 6.1.1.2.3

Rewrite the expression.

$x = \frac{6}{π} \arccos (- \frac{t}{15} + \frac{10}{3})$

Step 6.2

Simplify the right side.

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Step 6.2.1

Combine $\frac{6}{π}$ and $\arccos (- \frac{t}{15} + \frac{10}{3})$ .

$x = \frac{6 \arccos (- \frac{t}{15} + \frac{10}{3})}{π}$

Step 7

Simplify the numerator.

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Step 7.1

To write $\frac{10}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = \frac{6 \arccos (- \frac{t}{15} + \frac{10}{3} \cdot \frac{5}{5})}{π}$

Step 7.2

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.2.1

Multiply $\frac{10}{3}$ by $\frac{5}{5}$ .

$x = \frac{6 \arccos (- \frac{t}{15} + \frac{10 \cdot 5}{3 \cdot 5})}{π}$

Step 7.2.2

Multiply $3$ by $5$ .

$x = \frac{6 \arccos (- \frac{t}{15} + \frac{10 \cdot 5}{15})}{π}$

Step 7.3

Combine the numerators over the common denominator.

$x = \frac{6 \arccos (\frac{- t + 10 \cdot 5}{15})}{π}$

Step 7.4

Multiply $10$ by $5$ .

$x = \frac{6 \arccos (\frac{- t + 50}{15})}{π}$

Step 8

Multiply each term in $\frac{x π}{6} = \arccos (- \frac{t}{15} + \frac{10}{3})$ by $6$ to eliminate the fractions.

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Step 8.1

Multiply each term in $\frac{x π}{6} = \arccos (- \frac{t}{15} + \frac{10}{3})$ by $6$ .

$\frac{x π}{6} \cdot 6 = \arccos (- \frac{t}{15} + \frac{10}{3}) \cdot 6$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $6$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{x π}{6} \cdot 6 = \arccos (- \frac{t}{15} + \frac{10}{3}) \cdot 6$

Step 8.2.1.2

Rewrite the expression.

$x π = \arccos (- \frac{t}{15} + \frac{10}{3}) \cdot 6$

Step 8.3

Simplify the right side.

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Step 8.3.1

Move $6$ to the left of $\arccos (- \frac{t}{15} + \frac{10}{3})$ .

$x π = 6 \arccos (- \frac{t}{15} + \frac{10}{3})$

Step 9

Divide each term in $x π = 6 \arccos (- \frac{t}{15} + \frac{10}{3})$ by $π$ and simplify.

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Step 9.1

Divide each term in $x π = 6 \arccos (- \frac{t}{15} + \frac{10}{3})$ by $π$ .

$\frac{x π}{π} = \frac{6 \arccos (- \frac{t}{15} + \frac{10}{3})}{π}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $π$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{x π}{π} = \frac{6 \arccos (- \frac{t}{15} + \frac{10}{3})}{π}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{6 \arccos (- \frac{t}{15} + \frac{10}{3})}{π}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Simplify the numerator.

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Step 9.3.1.1

To write $\frac{10}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = \frac{6 \arccos (- \frac{t}{15} + \frac{10}{3} \cdot \frac{5}{5})}{π}$

Step 9.3.1.2

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.1.2.1

Multiply $\frac{10}{3}$ by $\frac{5}{5}$ .

$x = \frac{6 \arccos (- \frac{t}{15} + \frac{10 \cdot 5}{3 \cdot 5})}{π}$

Step 9.3.1.2.2

Multiply $3$ by $5$ .

$x = \frac{6 \arccos (- \frac{t}{15} + \frac{10 \cdot 5}{15})}{π}$

Step 9.3.1.3

Combine the numerators over the common denominator.

$x = \frac{6 \arccos (\frac{- t + 10 \cdot 5}{15})}{π}$

Step 9.3.1.4

Multiply $10$ by $5$ .

$x = \frac{6 \arccos (\frac{- t + 50}{15})}{π}$