Trigonometry Examples

$P (80) = \frac{90}{1 + 271 e^{- 0.122 \cdot 80}}$

Step 1

Simplify.

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Step 1.1

Multiply $- 0.122$ by $80$ .

$P (80) = \frac{90}{1 + 271 e^{- 9.76}}$

Step 1.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$P (80) = \frac{90}{1 + 271 \frac{1}{e^{9.76}}}$

Step 1.3

Combine $271$ and $\frac{1}{e^{9.76}}$ .

$P (80) = \frac{90}{1 + \frac{271}{e^{9.76}}}$

Step 1.4

Simplify the denominator.

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Step 1.4.1

Write $1$ as a fraction with a common denominator.

$P (80) = \frac{90}{\frac{e^{9.76}}{e^{9.76}} + \frac{271}{e^{9.76}}}$

Step 1.4.2

Combine the numerators over the common denominator.

$P (80) = \frac{90}{\frac{e^{9.76} + 271}{e^{9.76}}}$

Step 1.5

Multiply the numerator by the reciprocal of the denominator.

$P (80) = 90 \frac{e^{9.76}}{e^{9.76} + 271}$

Step 1.6

Combine $90$ and $\frac{e^{9.76}}{e^{9.76} + 271}$ .

$P (80) = \frac{90 e^{9.76}}{e^{9.76} + 271}$

Step 2

Divide each term in $P (80) = \frac{90 e^{9.76}}{e^{9.76} + 271}$ by $80$ and simplify.

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Step 2.1

Divide each term in $P (80) = \frac{90 e^{9.76}}{e^{9.76} + 271}$ by $80$ .

$\frac{P (80)}{80} = \frac{\frac{90 e^{9.76}}{e^{9.76} + 271}}{80}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $80$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{P \cdot 80}{80} = \frac{\frac{90 e^{9.76}}{e^{9.76} + 271}}{80}$

Step 2.2.1.2

Divide $P$ by $1$ .

$P = \frac{\frac{90 e^{9.76}}{e^{9.76} + 271}}{80}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply the numerator by the reciprocal of the denominator.

$P = \frac{90 e^{9.76}}{e^{9.76} + 271} \cdot \frac{1}{80}$

Step 2.3.2

Cancel the common factor of $10$ .

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Step 2.3.2.1

Factor $10$ out of $90 e^{9.76}$ .

$P = \frac{10 (9 e^{9.76})}{e^{9.76} + 271} \cdot \frac{1}{80}$

Step 2.3.2.2

Factor $10$ out of $80$ .

$P = \frac{10 (9 e^{9.76})}{e^{9.76} + 271} \cdot \frac{1}{10 (8)}$

Step 2.3.2.3

Cancel the common factor.

$P = \frac{10 (9 e^{9.76})}{e^{9.76} + 271} \cdot \frac{1}{10 \cdot 8}$

Step 2.3.2.4

Rewrite the expression.

$P = \frac{9 e^{9.76}}{e^{9.76} + 271} \cdot \frac{1}{8}$

Step 2.3.3

Multiply $\frac{9 e^{9.76}}{e^{9.76} + 271}$ by $\frac{1}{8}$ .

$P = \frac{9 e^{9.76}}{(e^{9.76} + 271) \cdot 8}$

Step 2.3.4

Move $8$ to the left of $e^{9.76} + 271$ .

$P = \frac{9 e^{9.76}}{8 (e^{9.76} + 271)}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$P = \frac{9 e^{9.76}}{8 (e^{9.76} + 271)}$

Decimal Form:

$P = 1.10767522 \dots$