Trigonometry Examples

$B = 127$ , $a = 32$ , $C = 25$

Step 1

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

$A + 25 + 127 = 180$

Step 2

Solve the equation for $A$ .

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Step 2.1

Add $25$ and $127$ .

$A + 152 = 180$

Step 2.2

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

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Step 2.2.1

Subtract $152$ from both sides of the equation.

$A = 180 - 152$

Step 2.2.2

Subtract $152$ from $180$ .

$A = 28$

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 $b$ .

$\frac{\sin (127)}{b} = \frac{\sin (28)}{32}$

Step 5

Solve the equation for $b$ .

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Step 5.1

Factor each term.

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Step 5.1.1

Evaluate $\sin (127)$ .

$\frac{0.79863551}{b} = \frac{\sin (28)}{32}$

Step 5.1.2

Evaluate $\sin (28)$ .

$\frac{0.79863551}{b} = \frac{0.46947156}{32}$

Step 5.1.3

Divide $0.46947156$ by $32$ .

$\frac{0.79863551}{b} = 0.01467098$

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.

$b, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$b$

Step 5.3

Multiply each term in $\frac{0.79863551}{b} = 0.01467098$ by $b$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $\frac{0.79863551}{b} = 0.01467098$ by $b$ .

$\frac{0.79863551}{b} b = 0.01467098 b$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $b$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{0.79863551}{b} b = 0.01467098 b$

Step 5.3.2.1.2

Rewrite the expression.

$0.79863551 = 0.01467098 b$

Step 5.4

Solve the equation.

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Step 5.4.1

Rewrite the equation as $0.01467098 b = 0.79863551$ .

$0.01467098 b = 0.79863551$

Step 5.4.2

Divide each term in $0.01467098 b = 0.79863551$ by $0.01467098$ and simplify.

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Step 5.4.2.1

Divide each term in $0.01467098 b = 0.79863551$ by $0.01467098$ .

$\frac{0.01467098 b}{0.01467098} = \frac{0.79863551}{0.01467098}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $0.01467098$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{0.01467098 b}{0.01467098} = \frac{0.79863551}{0.01467098}$

Step 5.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.79863551}{0.01467098}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Divide $0.79863551$ by $0.01467098$ .

$b = 54.43638837$

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 $c$ .

$\frac{\sin (25)}{c} = \frac{\sin (28)}{32}$

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 $\sin (25)$ .

$\frac{0.42261826}{c} = \frac{\sin (28)}{32}$

Step 8.1.2

Evaluate $\sin (28)$ .

$\frac{0.42261826}{c} = \frac{0.46947156}{32}$

Step 8.1.3

Divide $0.46947156$ by $32$ .

$\frac{0.42261826}{c} = 0.01467098$

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 $\frac{0.42261826}{c} = 0.01467098$ by $c$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.42261826}{c} = 0.01467098$ by $c$ .

$\frac{0.42261826}{c} c = 0.01467098 c$

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.

$\frac{0.42261826}{c} c = 0.01467098 c$

Step 8.3.2.1.2

Rewrite the expression.

$0.42261826 = 0.01467098 c$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.01467098 c = 0.42261826$ .

$0.01467098 c = 0.42261826$

Step 8.4.2

Divide each term in $0.01467098 c = 0.42261826$ by $0.01467098$ and simplify.

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Step 8.4.2.1

Divide each term in $0.01467098 c = 0.42261826$ by $0.01467098$ .

$\frac{0.01467098 c}{0.01467098} = \frac{0.42261826}{0.01467098}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.01467098$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.01467098 c}{0.01467098} = \frac{0.42261826}{0.01467098}$

Step 8.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.42261826}{0.01467098}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.42261826$ by $0.01467098$ .

$c = 28.80639733$

Step 9

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

$A = 28$

$B = 127$

$C = 25$

$a = 32$

$b = 54.43638837$

$c = 28.80639733$