Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = \\ a = & 105 \end{matrix} & \begin{matrix} A = & 65 \\ B = & 37 \\ C = \end{matrix} \end{matrix}$

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 $b$ .

$\frac{\sin (37)}{b} = \frac{\sin (65)}{105}$

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 $\sin (37)$ .

$\frac{0.60181502}{b} = \frac{\sin (65)}{105}$

Step 3.1.2

Evaluate $\sin (65)$ .

$\frac{0.60181502}{b} = \frac{0.90630778}{105}$

Step 3.1.3

Divide $0.90630778$ by $105$ .

$\frac{0.60181502}{b} = 0.0086315$

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 $\frac{0.60181502}{b} = 0.0086315$ by $b$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{0.60181502}{b} = 0.0086315$ by $b$ .

$\frac{0.60181502}{b} b = 0.0086315 b$

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.

$\frac{0.60181502}{b} b = 0.0086315 b$

Step 3.3.2.1.2

Rewrite the expression.

$0.60181502 = 0.0086315 b$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $0.0086315 b = 0.60181502$ .

$0.0086315 b = 0.60181502$

Step 3.4.2

Divide each term in $0.0086315 b = 0.60181502$ by $0.0086315$ and simplify.

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Step 3.4.2.1

Divide each term in $0.0086315 b = 0.60181502$ by $0.0086315$ .

$\frac{0.0086315 b}{0.0086315} = \frac{0.60181502}{0.0086315}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $0.0086315$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.0086315 b}{0.0086315} = \frac{0.60181502}{0.0086315}$

Step 3.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.60181502}{0.0086315}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $0.60181502$ by $0.0086315$ .

$b = 69.72308782$

Step 4

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

$65 + C + 37 = 180$

Step 5

Solve the equation for $C$ .

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Step 5.1

Add $65$ and $37$ .

$C + 102 = 180$

Step 5.2

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

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Step 5.2.1

Subtract $102$ from both sides of the equation.

$C = 180 - 102$

Step 5.2.2

Subtract $102$ from $180$ .

$C = 78$

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 $c$ .

$\frac{\sin (78)}{c} = \frac{\sin (65)}{105}$

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 $\sin (78)$ .

$\frac{0.9781476}{c} = \frac{\sin (65)}{105}$

Step 8.1.2

Evaluate $\sin (65)$ .

$\frac{0.9781476}{c} = \frac{0.90630778}{105}$

Step 8.1.3

Divide $0.90630778$ by $105$ .

$\frac{0.9781476}{c} = 0.0086315$

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 $\frac{0.9781476}{c} = 0.0086315$ by $c$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.9781476}{c} = 0.0086315$ by $c$ .

$\frac{0.9781476}{c} c = 0.0086315 c$

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.

$\frac{0.9781476}{c} c = 0.0086315 c$

Step 8.3.2.1.2

Rewrite the expression.

$0.9781476 = 0.0086315 c$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.0086315 c = 0.9781476$ .

$0.0086315 c = 0.9781476$

Step 8.4.2

Divide each term in $0.0086315 c = 0.9781476$ by $0.0086315$ and simplify.

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Step 8.4.2.1

Divide each term in $0.0086315 c = 0.9781476$ by $0.0086315$ .

$\frac{0.0086315 c}{0.0086315} = \frac{0.9781476}{0.0086315}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.0086315$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.0086315 c}{0.0086315} = \frac{0.9781476}{0.0086315}$

Step 8.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.9781476}{0.0086315}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.9781476$ by $0.0086315$ .

$c = 113.32297873$

Step 9

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

$A = 65$

$B = 37$

$C = 78$

$a = 105$

$b = 69.72308782$

$c = 113.32297873$