Trigonometry Examples

$\frac{a}{\sin (46)} = \frac{10}{\sin (B)}$

Step 1

Rewrite the equation as $\frac{10}{\sin (B)} = \frac{a}{\sin (46)}$ .

$\frac{10}{\sin (B)} = \frac{a}{\sin (46)}$

Step 2

Factor each term.

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Step 2.1

Separate fractions.

$\frac{10}{\sin (B)} = \frac{a}{1} \cdot \frac{1}{\sin (46)}$

Step 2.2

Convert from $\frac{1}{\sin (46)}$ to $\csc (46)$ .

$\frac{10}{\sin (B)} = \frac{a}{1} \csc (46)$

Step 2.3

Divide $a$ by $1$ .

$\frac{10}{\sin (B)} = a \csc (46)$

Step 2.4

Evaluate $\csc (46)$ .

$\frac{10}{\sin (B)} = a \cdot 1.39016359$

Step 2.5

Move $1.39016359$ to the left of $a$ .

$\frac{10}{\sin (B)} = 1.39016359 a$

Step 3

Find the LCD of the terms in the equation.

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Step 3.1

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

$\sin (B), 1$

Step 3.2

The LCM of one and any expression is the expression.

$\sin (B)$

Step 4

Multiply each term in $\frac{10}{\sin (B)} = 1.39016359 a$ by $\sin (B)$ to eliminate the fractions.

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Step 4.1

Multiply each term in $\frac{10}{\sin (B)} = 1.39016359 a$ by $\sin (B)$ .

$\frac{10}{\sin (B)} \sin (B) = 1.39016359 a \sin (B)$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $\sin (B)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{10}{\sin (B)} \sin (B) = 1.39016359 a \sin (B)$

Step 4.2.1.2

Rewrite the expression.

$10 = 1.39016359 a \sin (B)$

Step 5

Solve the equation.

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Step 5.1

Rewrite the equation as $1.39016359 a \sin (B) = 10$ .

$1.39016359 a \sin (B) = 10$

Step 5.2

Divide each term in $1.39016359 a \sin (B) = 10$ by $1.39016359 a$ and simplify.

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Step 5.2.1

Divide each term in $1.39016359 a \sin (B) = 10$ by $1.39016359 a$ .

$\frac{1.39016359 a \sin (B)}{1.39016359 a} = \frac{10}{1.39016359 a}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $1.39016359$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{1.39016359 a \sin (B)}{1.39016359 a} = \frac{10}{1.39016359 a}$

Step 5.2.2.1.2

Rewrite the expression.

$\frac{a \sin (B)}{a} = \frac{10}{1.39016359 a}$

Step 5.2.2.2

Cancel the common factor of $a$ .

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Step 5.2.2.2.1

Cancel the common factor.

$\frac{a \sin (B)}{a} = \frac{10}{1.39016359 a}$

Step 5.2.2.2.2

Divide $\sin (B)$ by $1$ .

$\sin (B) = \frac{10}{1.39016359 a}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Factor $10$ out of $10$ .

$\sin (B) = \frac{10 (1)}{1.39016359 a}$

Step 5.2.3.2

Factor $1.39016359$ out of $1.39016359 a$ .

$\sin (B) = \frac{10 (1)}{1.39016359 (a)}$

Step 5.2.3.3

Separate fractions.

$\sin (B) = \frac{10}{1.39016359} \cdot \frac{1}{a}$

Step 5.2.3.4

Divide $10$ by $1.39016359$ .

$\sin (B) = 7.193398 \frac{1}{a}$

Step 5.2.3.5

Combine $7.193398$ and $\frac{1}{a}$ .

$\sin (B) = \frac{7.193398}{a}$

Step 6

Take the inverse sine of both sides of the equation to extract $B$ from inside the sine.

$B = \arcsin (\frac{7.193398}{a})$