Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 33.2 \\ c = \\ a = \end{matrix} & \begin{matrix} A = \\ B = & 61 ° \\ C = & 90 ° \end{matrix} \end{matrix}$

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 $c$ .

$\frac{\sin (90 °)}{c} = \frac{\sin (61 °)}{33.2}$

Step 3

Solve the equation for $c$ .

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Step 3.1

Factor each term.

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Step 3.1.1

The exact value of $\sin (90 °)$ is $1$ .

$\frac{1}{c} = \frac{\sin (61 °)}{33.2}$

Step 3.1.2

Evaluate $\sin (61 °)$ .

$\frac{1}{c} = \frac{0.8746197}{33.2}$

Step 3.1.3

Divide $0.8746197$ by $33.2$ .

$\frac{1}{c} = 0.02634396$

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.

$c, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$c$

Step 3.3

Multiply each term in $\frac{1}{c} = 0.02634396$ by $c$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{1}{c} = 0.02634396$ by $c$ .

$\frac{1}{c} c = 0.02634396 c$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $c$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{1}{c} c = 0.02634396 c$

Step 3.3.2.1.2

Rewrite the expression.

$1 = 0.02634396 c$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $0.02634396 c = 1$ .

$0.02634396 c = 1$

Step 3.4.2

Divide each term in $0.02634396 c = 1$ by $0.02634396$ and simplify.

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Step 3.4.2.1

Divide each term in $0.02634396 c = 1$ by $0.02634396$ .

$\frac{0.02634396 c}{0.02634396} = \frac{1}{0.02634396}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $0.02634396$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.02634396 c}{0.02634396} = \frac{1}{0.02634396}$

Step 3.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{1}{0.02634396}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $1$ by $0.02634396$ .

$c = 37.95935505$

Step 4

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

$A + 90 ° + 61 ° = 180$

Step 5

Solve the equation for $A$ .

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Step 5.1

Add $90 °$ and $61 °$ .

$A + 151 = 180$

Step 5.2

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

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Step 5.2.1

Subtract $151$ from both sides of the equation.

$A = 180 - 151$

Step 5.2.2

Subtract $151$ from $180$ .

$A = 29$

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 $a$ .

$\frac{\sin (29)}{a} = \frac{\sin (61 °)}{33.2}$

Step 8

Solve the equation for $a$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (29)$ .

$\frac{0.48480962}{a} = \frac{\sin (61 °)}{33.2}$

Step 8.1.2

Evaluate $\sin (61 °)$ .

$\frac{0.48480962}{a} = \frac{0.8746197}{33.2}$

Step 8.1.3

Divide $0.8746197$ by $33.2$ .

$\frac{0.48480962}{a} = 0.02634396$

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.

$a, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$a$

Step 8.3

Multiply each term in $\frac{0.48480962}{a} = 0.02634396$ by $a$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.48480962}{a} = 0.02634396$ by $a$ .

$\frac{0.48480962}{a} a = 0.02634396 a$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $a$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.48480962}{a} a = 0.02634396 a$

Step 8.3.2.1.2

Rewrite the expression.

$0.48480962 = 0.02634396 a$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.02634396 a = 0.48480962$ .

$0.02634396 a = 0.48480962$

Step 8.4.2

Divide each term in $0.02634396 a = 0.48480962$ by $0.02634396$ and simplify.

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Step 8.4.2.1

Divide each term in $0.02634396 a = 0.48480962$ by $0.02634396$ .

$\frac{0.02634396 a}{0.02634396} = \frac{0.48480962}{0.02634396}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.02634396$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.02634396 a}{0.02634396} = \frac{0.48480962}{0.02634396}$

Step 8.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0.48480962}{0.02634396}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.48480962$ by $0.02634396$ .

$a = 18.4030605$

Step 9

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

$A = 29$

$B = 61 °$

$C = 90 °$

$a = 18.4030605$

$b = 33.2$

$c = 37.95935505$