Trigonometry Examples

$A = 53$ , $B = 23$ , $C = 104$ , $a = \frac{29}{8}$

Step 1

The law of sines is based on the proportionality of sides and angles in triangles. The law states that for the angles of a non-right triangle, each angle of the triangle has the same ratio of angle measure to sine value.

$\frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c}$

Step 2

Substitute the known values into the law of sines to find $b$ .

$\frac{\sin (23)}{b} = \frac{\frac{\sin (53)}{29}}{8}$

Step 3

Solve the equation for $b$ .

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Step 3.1

Factor each term.

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Step 3.1.1

Evaluate $\sin (23)$ .

$\frac{0.39073112}{b} = \frac{\frac{\sin (53)}{29}}{8}$

Step 3.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.39073112}{b} = \frac{\sin (53)}{29} \cdot \frac{1}{8}$

Step 3.1.3

Evaluate $\sin (53)$ .

$\frac{0.39073112}{b} = \frac{0.79863551}{29} \cdot \frac{1}{8}$

Step 3.1.4

Divide $0.79863551$ by $29$ .

$\frac{0.39073112}{b} = 0.02753915 (\frac{1}{8})$

Step 3.1.5

Combine $0.02753915$ and $\frac{1}{8}$ .

$\frac{0.39073112}{b} = \frac{0.02753915}{8}$

Step 3.1.6

Divide $0.02753915$ by $8$ .

$\frac{0.39073112}{b} = 0.00344239$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

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

$b, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$b$

Step 3.3

Multiply each term in $\frac{0.39073112}{b} = 0.00344239$ by $b$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{0.39073112}{b} = 0.00344239$ by $b$ .

$\frac{0.39073112}{b} b = 0.00344239 b$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $b$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{0.39073112}{b} b = 0.00344239 b$

Step 3.3.2.1.2

Rewrite the expression.

$0.39073112 = 0.00344239 b$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $0.00344239 b = 0.39073112$ .

$0.00344239 b = 0.39073112$

Step 3.4.2

Divide each term in $0.00344239 b = 0.39073112$ by $0.00344239$ and simplify.

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Step 3.4.2.1

Divide each term in $0.00344239 b = 0.39073112$ by $0.00344239$ .

$\frac{0.00344239 b}{0.00344239} = \frac{0.39073112}{0.00344239}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $0.00344239$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.00344239 b}{0.00344239} = \frac{0.39073112}{0.00344239}$

Step 3.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.39073112}{0.00344239}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $0.39073112$ by $0.00344239$ .

$b = 113.50562386$

Step 4

$\frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c}$

Step 5

Substitute the known values into the law of sines to find $c$ .

$\frac{\sin (104)}{c} = \frac{\frac{\sin (53)}{29}}{8}$

Step 6

Solve the equation for $c$ .

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Step 6.1

Factor each term.

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Step 6.1.1

Evaluate $\sin (104)$ .

$\frac{0.97029572}{c} = \frac{\frac{\sin (53)}{29}}{8}$

Step 6.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.97029572}{c} = \frac{\sin (53)}{29} \cdot \frac{1}{8}$

Step 6.1.3

Evaluate $\sin (53)$ .

$\frac{0.97029572}{c} = \frac{0.79863551}{29} \cdot \frac{1}{8}$

Step 6.1.4

Divide $0.79863551$ by $29$ .

$\frac{0.97029572}{c} = 0.02753915 (\frac{1}{8})$

Step 6.1.5

Combine $0.02753915$ and $\frac{1}{8}$ .

$\frac{0.97029572}{c} = \frac{0.02753915}{8}$

Step 6.1.6

Divide $0.02753915$ by $8$ .

$\frac{0.97029572}{c} = 0.00344239$

Step 6.2

Find the LCD of the terms in the equation.

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Step 6.2.1

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

$c, 1$

Step 6.2.2

The LCM of one and any expression is the expression.

$c$

Step 6.3

Multiply each term in $\frac{0.97029572}{c} = 0.00344239$ by $c$ to eliminate the fractions.

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Step 6.3.1

Multiply each term in $\frac{0.97029572}{c} = 0.00344239$ by $c$ .

$\frac{0.97029572}{c} c = 0.00344239 c$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $c$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{0.97029572}{c} c = 0.00344239 c$

Step 6.3.2.1.2

Rewrite the expression.

$0.97029572 = 0.00344239 c$

Step 6.4

Solve the equation.

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Step 6.4.1

Rewrite the equation as $0.00344239 c = 0.97029572$ .

$0.00344239 c = 0.97029572$

Step 6.4.2

Divide each term in $0.00344239 c = 0.97029572$ by $0.00344239$ and simplify.

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Step 6.4.2.1

Divide each term in $0.00344239 c = 0.97029572$ by $0.00344239$ .

$\frac{0.00344239 c}{0.00344239} = \frac{0.97029572}{0.00344239}$

Step 6.4.2.2

Simplify the left side.

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Step 6.4.2.2.1

Cancel the common factor of $0.00344239$ .

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Step 6.4.2.2.1.1

Cancel the common factor.

$\frac{0.00344239 c}{0.00344239} = \frac{0.97029572}{0.00344239}$

Step 6.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.97029572}{0.00344239}$

Step 6.4.2.3

Simplify the right side.

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Step 6.4.2.3.1

Divide $0.97029572$ by $0.00344239$ .

$c = 281.86651565$

Step 7

These are the results for all angles and sides for the given triangle.

$A = 53$

$B = 23$

$C = 104$

$a = \frac{29}{8}$

$b = 113.50562386$

$c = 281.86651565$