Trigonometry Examples

$A = 102.4$ , $a = 21.6$ , $C = 16.7$ , $B = 60.9$

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 $b$ .

$\frac{\sin (60.9)}{b} = \frac{\sin (102.4)}{21.6}$

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 $\sin (60.9)$ .

$\frac{0.87377222}{b} = \frac{\sin (102.4)}{21.6}$

Step 3.1.2

Evaluate $\sin (102.4)$ .

$\frac{0.87377222}{b} = \frac{0.97667227}{21.6}$

Step 3.1.3

Divide $0.97667227$ by $21.6$ .

$\frac{0.87377222}{b} = 0.0452163$

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 $\frac{0.87377222}{b} = 0.0452163$ by $b$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{0.87377222}{b} = 0.0452163$ by $b$ .

$\frac{0.87377222}{b} b = 0.0452163 b$

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.

$\frac{0.87377222}{b} b = 0.0452163 b$

Step 3.3.2.1.2

Rewrite the expression.

$0.87377222 = 0.0452163 b$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $0.0452163 b = 0.87377222$ .

$0.0452163 b = 0.87377222$

Step 3.4.2

Divide each term in $0.0452163 b = 0.87377222$ by $0.0452163$ and simplify.

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Step 3.4.2.1

Divide each term in $0.0452163 b = 0.87377222$ by $0.0452163$ .

$\frac{0.0452163 b}{0.0452163} = \frac{0.87377222}{0.0452163}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $0.0452163$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.0452163 b}{0.0452163} = \frac{0.87377222}{0.0452163}$

Step 3.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.87377222}{0.0452163}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $0.87377222$ by $0.0452163$ .

$b = 19.32427123$

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 $c$ .

$\frac{\sin (16.7)}{c} = \frac{\sin (102.4)}{21.6}$

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 $\sin (16.7)$ .

$\frac{0.28736051}{c} = \frac{\sin (102.4)}{21.6}$

Step 6.1.2

Evaluate $\sin (102.4)$ .

$\frac{0.28736051}{c} = \frac{0.97667227}{21.6}$

Step 6.1.3

Divide $0.97667227$ by $21.6$ .

$\frac{0.28736051}{c} = 0.0452163$

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 $\frac{0.28736051}{c} = 0.0452163$ by $c$ to eliminate the fractions.

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Step 6.3.1

Multiply each term in $\frac{0.28736051}{c} = 0.0452163$ by $c$ .

$\frac{0.28736051}{c} c = 0.0452163 c$

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.

$\frac{0.28736051}{c} c = 0.0452163 c$

Step 6.3.2.1.2

Rewrite the expression.

$0.28736051 = 0.0452163 c$

Step 6.4

Solve the equation.

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Step 6.4.1

Rewrite the equation as $0.0452163 c = 0.28736051$ .

$0.0452163 c = 0.28736051$

Step 6.4.2

Divide each term in $0.0452163 c = 0.28736051$ by $0.0452163$ and simplify.

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Step 6.4.2.1

Divide each term in $0.0452163 c = 0.28736051$ by $0.0452163$ .

$\frac{0.0452163 c}{0.0452163} = \frac{0.28736051}{0.0452163}$

Step 6.4.2.2

Simplify the left side.

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Step 6.4.2.2.1

Cancel the common factor of $0.0452163$ .

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Step 6.4.2.2.1.1

Cancel the common factor.

$\frac{0.0452163 c}{0.0452163} = \frac{0.28736051}{0.0452163}$

Step 6.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.28736051}{0.0452163}$

Step 6.4.2.3

Simplify the right side.

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Step 6.4.2.3.1

Divide $0.28736051$ by $0.0452163$ .

$c = 6.35524051$

Step 7

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

$A = 102.4$

$B = 60.9$

$C = 16.7$

$a = 21.6$

$b = 19.32427123$

$c = 6.35524051$