Trigonometry Examples

$f (x) = \ln (\arcsin (\frac{x + 2}{5 - x}))$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\arcsin (\frac{x + 2}{5 - x}) = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

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

$\frac{x + 2}{5 - x} = \sin (0)$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Split the fraction $\frac{x + 2}{5 - x}$ into two fractions.

$\frac{x}{5 - x} + \frac{2}{5 - x} = \sin (0)$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

The exact value of $\sin (0)$ is $0$ .

$\frac{x}{5 - x} + \frac{2}{5 - x} = 0$

Step 1.2.4

Find the LCD of the terms in the equation.

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Step 1.2.4.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$5 - x, 5 - x, 1$

Step 1.2.4.2

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.2.4.3

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.2.4.4

The LCM of $1, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

Step 1.2.4.5

The factor for $5 - x$ is $5 - x$ itself.

$(5 - x) = 5 - x$

$(5 - x)$ occurs $1$ time.

Step 1.2.4.6

The LCM of $5 - x, 5 - x$ is the result of multiplying all factors the greatest number of times they occur in either term.

$5 - x$

Step 1.2.5

Multiply each term in $\frac{x}{5 - x} + \frac{2}{5 - x} = 0$ by $5 - x$ to eliminate the fractions.

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Step 1.2.5.1

Multiply each term in $\frac{x}{5 - x} + \frac{2}{5 - x} = 0$ by $5 - x$ .

$\frac{x}{5 - x} (5 - x) + \frac{2}{5 - x} (5 - x) = 0 (5 - x)$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Cancel the common factor of $5 - x$ .

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Step 1.2.5.2.1.1.1

Cancel the common factor.

$\frac{x}{5 - x} (5 - x) + \frac{2}{5 - x} (5 - x) = 0 (5 - x)$

Step 1.2.5.2.1.1.2

Rewrite the expression.

$x + \frac{2}{5 - x} (5 - x) = 0 (5 - x)$

Step 1.2.5.2.1.2

Cancel the common factor of $5 - x$ .

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Step 1.2.5.2.1.2.1

Cancel the common factor.

$x + \frac{2}{5 - x} (5 - x) = 0 (5 - x)$

Step 1.2.5.2.1.2.2

Rewrite the expression.

$x + 2 = 0 (5 - x)$

Step 1.2.5.3

Simplify the right side.

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Step 1.2.5.3.1

Multiply $0$ by $5 - x$ .

$x + 2 = 0$

Step 1.2.6

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.3

The vertical asymptote occurs at $x = - 2$ .

Vertical Asymptote: $x = - 2$

Step 2

Find the point at $x = - 1$ .

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Step 2.1

Replace the variable $x$ with $- 1$ in the expression.

$f (- 1) = \ln (\arcsin (\frac{(- 1) + 2}{5 - (- 1)}))$

Step 2.2

Simplify the result.

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Step 2.2.1

Add $- 1$ and $2$ .

$f (- 1) = \ln (\arcsin (\frac{1}{5 - (- 1)}))$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

Multiply $- 1$ by $- 1$ .

$f (- 1) = \ln (\arcsin (\frac{1}{5 + 1}))$

Step 2.2.2.2

Add $5$ and $1$ .

$f (- 1) = \ln (\arcsin (\frac{1}{6}))$

Step 2.2.3

Evaluate $\arcsin (\frac{1}{6})$ .

$f (- 1) = \ln (0.16744807)$

Step 2.2.4

The final answer is $\ln (0.16744807)$ .

$\ln (0.16744807)$

Step 3

Find the point at $x = 0$ .

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Step 3.1

Replace the variable $x$ with $0$ in the expression.

$f (0) = \ln (\arcsin (\frac{(0) + 2}{5 - (0)}))$

Step 3.2

Simplify the result.

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Step 3.2.1

Add $0$ and $2$ .

$f (0) = \ln (\arcsin (\frac{2}{5 - (0)}))$

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

Multiply $- 1$ by $0$ .

$f (0) = \ln (\arcsin (\frac{2}{5 + 0}))$

Step 3.2.2.2

Add $5$ and $0$ .

$f (0) = \ln (\arcsin (\frac{2}{5}))$

Step 3.2.3

Evaluate $\arcsin (\frac{2}{5})$ .

$f (0) = \ln (0.41151684)$

Step 3.2.4

The final answer is $\ln (0.41151684)$ .

$\ln (0.41151684)$

Step 4

Find the point at $x = 1$ .

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Step 4.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \ln (\arcsin (\frac{(1) + 2}{5 - (1)}))$

Step 4.2

Simplify the result.

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Step 4.2.1

Add $1$ and $2$ .

$f (1) = \ln (\arcsin (\frac{3}{5 - (1)}))$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

Multiply $- 1$ by $1$ .

$f (1) = \ln (\arcsin (\frac{3}{5 - 1}))$

Step 4.2.2.2

Subtract $1$ from $5$ .

$f (1) = \ln (\arcsin (\frac{3}{4}))$

Step 4.2.3

Evaluate $\arcsin (\frac{3}{4})$ .

$f (1) = \ln (0.84806207)$

Step 4.2.4

The final answer is $\ln (0.84806207)$ .

$\ln (0.84806207)$

Step 5

The log function can be graphed using the vertical asymptote at $x = - 2$ and the points $(- 1, - 1.78708195), (0, - 0.88790532), (1, - 0.16480143)$ .

Vertical Asymptote: $x = - 2$

$\begin{array}{c} x & y \\ - 1 & - 1.787 \\ 0 & - 0.888 \\ 1 & - 0.165 \end{array}$

Step 6