Trigonometry Examples

$A = 131$ , $C = 23$ , $b = 10$

Step 1

The sum of all the angles in a triangle is $180$ degrees.

$131 + 23 + B = 180$

Step 2

Solve the equation for $B$ .

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Step 2.1

Add $131$ and $23$ .

$154 + B = 180$

Step 2.2

Move all terms not containing $B$ to the right side of the equation.

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Step 2.2.1

Subtract $154$ from both sides of the equation.

$B = 180 - 154$

Step 2.2.2

Subtract $154$ from $180$ .

$B = 26$

Step 3

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 4

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

$\frac{\sin (131)}{a} = \frac{\sin (26)}{10}$

Step 5

Solve the equation for $a$ .

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Step 5.1

Factor each term.

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Step 5.1.1

Evaluate $\sin (131)$ .

$\frac{0.75470958}{a} = \frac{\sin (26)}{10}$

Step 5.1.2

Evaluate $\sin (26)$ .

$\frac{0.75470958}{a} = \frac{0.43837114}{10}$

Step 5.1.3

Divide $0.43837114$ by $10$ .

$\frac{0.75470958}{a} = 0.04383711$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

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

$a, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$a$

Step 5.3

Multiply each term in $\frac{0.75470958}{a} = 0.04383711$ by $a$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $\frac{0.75470958}{a} = 0.04383711$ by $a$ .

$\frac{0.75470958}{a} a = 0.04383711 a$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $a$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{0.75470958}{a} a = 0.04383711 a$

Step 5.3.2.1.2

Rewrite the expression.

$0.75470958 = 0.04383711 a$

Step 5.4

Solve the equation.

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Step 5.4.1

Rewrite the equation as $0.04383711 a = 0.75470958$ .

$0.04383711 a = 0.75470958$

Step 5.4.2

Divide each term in $0.04383711 a = 0.75470958$ by $0.04383711$ and simplify.

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Step 5.4.2.1

Divide each term in $0.04383711 a = 0.75470958$ by $0.04383711$ .

$\frac{0.04383711 a}{0.04383711} = \frac{0.75470958}{0.04383711}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $0.04383711$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{0.04383711 a}{0.04383711} = \frac{0.75470958}{0.04383711}$

Step 5.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0.75470958}{0.04383711}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Divide $0.75470958$ by $0.04383711$ .

$a = 17.21622387$

Step 6

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

Step 7

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

$\frac{\sin (23)}{c} = \frac{\sin (131)}{17.21622387}$

Step 8

Solve the equation for $c$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (23)$ .

$\frac{0.39073112}{c} = \frac{\sin (131)}{17.21622387}$

Step 8.1.2

Evaluate $\sin (131)$ .

$\frac{0.39073112}{c} = \frac{0.75470958}{17.21622387}$

Step 8.1.3

Divide $0.75470958$ by $17.21622387$ .

$\frac{0.39073112}{c} = 0.04383711$

Step 8.2

Find the LCD of the terms in the equation.

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Step 8.2.1

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

$c, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$c$

Step 8.3

Multiply each term in $\frac{0.39073112}{c} = 0.04383711$ by $c$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.39073112}{c} = 0.04383711$ by $c$ .

$\frac{0.39073112}{c} c = 0.04383711 c$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $c$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.39073112}{c} c = 0.04383711 c$

Step 8.3.2.1.2

Rewrite the expression.

$0.39073112 = 0.04383711 c$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.04383711 c = 0.39073112$ .

$0.04383711 c = 0.39073112$

Step 8.4.2

Divide each term in $0.04383711 c = 0.39073112$ by $0.04383711$ and simplify.

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Step 8.4.2.1

Divide each term in $0.04383711 c = 0.39073112$ by $0.04383711$ .

$\frac{0.04383711 c}{0.04383711} = \frac{0.39073112}{0.04383711}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.04383711$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.04383711 c}{0.04383711} = \frac{0.39073112}{0.04383711}$

Step 8.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.39073112}{0.04383711}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.39073112$ by $0.04383711$ .

$c = 8.91324922$

Step 9

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

$A = 131$

$B = 26$

$C = 23$

$a = 17.21622387$

$b = 10$

$c = 8.91324922$