Trigonometry Examples

$c = 26$ , $a = 52$ , $A = 127$

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 $C$ .

$\frac{\sin (C)}{26} = \frac{\sin (127)}{52}$

Step 3

Solve the equation for $C$ .

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Step 3.1

Multiply both sides of the equation by $26$ .

$26 \frac{\sin (C)}{26} = 26 \frac{\sin (127)}{52}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $26$ .

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Step 3.2.1.1.1

Cancel the common factor.

$26 \frac{\sin (C)}{26} = 26 \frac{\sin (127)}{52}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (C) = 26 \frac{\sin (127)}{52}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $26 \frac{\sin (127)}{52}$ .

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Step 3.2.2.1.1

Cancel the common factor of $26$ .

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Step 3.2.2.1.1.1

Factor $26$ out of $52$ .

$\sin (C) = 26 \frac{\sin (127)}{26 (2)}$

Step 3.2.2.1.1.2

Cancel the common factor.

$\sin (C) = 26 \frac{\sin (127)}{26 \cdot 2}$

Step 3.2.2.1.1.3

Rewrite the expression.

$\sin (C) = \frac{\sin (127)}{2}$

Step 3.2.2.1.2

Evaluate $\sin (127)$ .

$\sin (C) = \frac{0.79863551}{2}$

Step 3.2.2.1.3

Divide $0.79863551$ by $2$ .

$\sin (C) = 0.39931775$

Step 3.3

Take the inverse sine of both sides of the equation to extract $C$ from inside the sine.

$C = \arcsin (0.39931775)$

Step 3.4

Simplify the right side.

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Step 3.4.1

Evaluate $\arcsin (0.39931775)$ .

$C = 23.53553497$

Step 3.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $180$ to find the solution in the second quadrant.

$C = 180 - 23.53553497$

Step 3.6

Subtract $23.53553497$ from $180$ .

$C = 156.46446502$

Step 3.7

The solution to the equation $C = 23.53553497$ .

$C = 23.53553497, 156.46446502$

Step 3.8

Exclude the invalid angle.

$C = 23.53553497$

Step 4

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

$127 + 23.53553497 + B = 180$

Step 5

Solve the equation for $B$ .

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Step 5.1

Add $127$ and $23.53553497$ .

$150.53553497 + B = 180$

Step 5.2

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

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Step 5.2.1

Subtract $150.53553497$ from both sides of the equation.

$B = 180 - 150.53553497$

Step 5.2.2

Subtract $150.53553497$ from $180$ .

$B = 29.46446502$

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 $b$ .

$\frac{\sin (29.46446502)}{b} = \frac{\sin (127)}{52}$

Step 8

Solve the equation for $b$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (29.46446502)$ .

$\frac{0.49188366}{b} = \frac{\sin (127)}{52}$

Step 8.1.2

Evaluate $\sin (127)$ .

$\frac{0.49188366}{b} = \frac{0.79863551}{52}$

Step 8.1.3

Divide $0.79863551$ by $52$ .

$\frac{0.49188366}{b} = 0.01535837$

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.

$b, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$b$

Step 8.3

Multiply each term in $\frac{0.49188366}{b} = 0.01535837$ by $b$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.49188366}{b} = 0.01535837$ by $b$ .

$\frac{0.49188366}{b} b = 0.01535837 b$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $b$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.49188366}{b} b = 0.01535837 b$

Step 8.3.2.1.2

Rewrite the expression.

$0.49188366 = 0.01535837 b$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.01535837 b = 0.49188366$ .

$0.01535837 b = 0.49188366$

Step 8.4.2

Divide each term in $0.01535837 b = 0.49188366$ by $0.01535837$ and simplify.

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Step 8.4.2.1

Divide each term in $0.01535837 b = 0.49188366$ by $0.01535837$ .

$\frac{0.01535837 b}{0.01535837} = \frac{0.49188366}{0.01535837}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.01535837$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.01535837 b}{0.01535837} = \frac{0.49188366}{0.01535837}$

Step 8.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.49188366}{0.01535837}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.49188366$ by $0.01535837$ .

$b = 32.02706422$

Step 9

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

$A = 127$

$B = 29.46446502$

$C = 23.53553497$

$a = 52$

$b = 32.02706422$

$c = 26$