Trigonometry Examples

$P = 100 - 20 \cos (\frac{8 π t}{3})$

Step 1

Rewrite the equation as $100 - 20 \cos (\frac{8 π t}{3}) = P$ .

$100 - 20 \cos (\frac{8 π t}{3}) = P$

Step 2

Subtract $100$ from both sides of the equation.

$- 20 \cos (\frac{8 π t}{3}) = P - 100$

Step 3

Divide each term in $- 20 \cos (\frac{8 π t}{3}) = P - 100$ by $- 20$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $- 20 \cos (\frac{8 π t}{3}) = P - 100$ by $- 20$ .

$\frac{- 20 \cos (\frac{8 π t}{3})}{- 20} = \frac{P}{- 20} + \frac{- 100}{- 20}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $- 20$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{- 20 \cos (\frac{8 π t}{3})}{- 20} = \frac{P}{- 20} + \frac{- 100}{- 20}$

Step 3.2.1.2

Divide $\cos (\frac{8 π t}{3})$ by $1$ .

$\cos (\frac{8 π t}{3}) = \frac{P}{- 20} + \frac{- 100}{- 20}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Simplify each term.

Tap for more steps...

Step 3.3.1.1

Move the negative in front of the fraction.

$\cos (\frac{8 π t}{3}) = - \frac{P}{20} + \frac{- 100}{- 20}$

Step 3.3.1.2

Divide $- 100$ by $- 20$ .

$\cos (\frac{8 π t}{3}) = - \frac{P}{20} + 5$

Step 4

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

$\frac{8 π t}{3} = \arccos (- \frac{P}{20} + 5)$

Step 5

Multiply both sides of the equation by $\frac{3}{8 π}$ .

$\frac{3}{8 π} \cdot \frac{8 π t}{3} = \frac{3}{8 π} \arccos (- \frac{P}{20} + 5)$

Step 6

Simplify both sides of the equation.

Tap for more steps...

Step 6.1

Simplify the left side.

Tap for more steps...

Step 6.1.1

Simplify $\frac{3}{8 π} \cdot \frac{8 π t}{3}$ .

Tap for more steps...

Step 6.1.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.1.1.1.1

Cancel the common factor.

$\frac{3}{8 π} \cdot \frac{8 π t}{3} = \frac{3}{8 π} \arccos (- \frac{P}{20} + 5)$

Step 6.1.1.1.2

Rewrite the expression.

$\frac{1}{8 π} (8 π t) = \frac{3}{8 π} \arccos (- \frac{P}{20} + 5)$

Step 6.1.1.2

Cancel the common factor of $8 π$ .

Tap for more steps...

Step 6.1.1.2.1

Factor $8 π$ out of $8 π t$ .

$\frac{1}{8 π} (8 π (t)) = \frac{3}{8 π} \arccos (- \frac{P}{20} + 5)$

Step 6.1.1.2.2

Cancel the common factor.

$\frac{1}{8 π} (8 π t) = \frac{3}{8 π} \arccos (- \frac{P}{20} + 5)$

Step 6.1.1.2.3

Rewrite the expression.

$t = \frac{3}{8 π} \arccos (- \frac{P}{20} + 5)$

Step 6.2

Simplify the right side.

Tap for more steps...

Step 6.2.1

Combine $\frac{3}{8 π}$ and $\arccos (- \frac{P}{20} + 5)$ .

$t = \frac{3 \arccos (- \frac{P}{20} + 5)}{8 π}$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{20}{20}$ .

$t = \frac{3 \arccos (- \frac{P}{20} + 5 \cdot \frac{20}{20})}{8 π}$

Step 7.2

Combine $5$ and $\frac{20}{20}$ .

$t = \frac{3 \arccos (- \frac{P}{20} + \frac{5 \cdot 20}{20})}{8 π}$

Step 7.3

Combine the numerators over the common denominator.

$t = \frac{3 \arccos (\frac{- P + 5 \cdot 20}{20})}{8 π}$

Step 7.4

Multiply $5$ by $20$ .

$t = \frac{3 \arccos (\frac{- P + 100}{20})}{8 π}$