Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 54 \\ c = & 42 \\ a = \end{matrix} & \begin{matrix} A = \\ B = & 52.1 \\ C = \end{matrix} \end{matrix}$

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 $C$ .

$\frac{\sin (C)}{42} = \frac{\sin (52.1)}{54}$

Step 3

Solve the equation for $C$ .

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Step 3.1

Multiply both sides of the equation by $42$ .

$42 \frac{\sin (C)}{42} = 42 \frac{\sin (52.1)}{54}$

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 $42$ .

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Step 3.2.1.1.1

Cancel the common factor.

$42 \frac{\sin (C)}{42} = 42 \frac{\sin (52.1)}{54}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (C) = 42 \frac{\sin (52.1)}{54}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $42 \frac{\sin (52.1)}{54}$ .

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Step 3.2.2.1.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1.1

Factor $6$ out of $42$ .

$\sin (C) = 6 (7) \frac{\sin (52.1)}{54}$

Step 3.2.2.1.1.2

Factor $6$ out of $54$ .

$\sin (C) = 6 \cdot 7 \frac{\sin (52.1)}{6 \cdot 9}$

Step 3.2.2.1.1.3

Cancel the common factor.

$\sin (C) = 6 \cdot 7 \frac{\sin (52.1)}{6 \cdot 9}$

Step 3.2.2.1.1.4

Rewrite the expression.

$\sin (C) = 7 \frac{\sin (52.1)}{9}$

Step 3.2.2.1.2

Combine $7$ and $\frac{\sin (52.1)}{9}$ .

$\sin (C) = \frac{7 \sin (52.1)}{9}$

Step 3.2.2.1.3

Evaluate $\sin (52.1)$ .

$\sin (C) = \frac{7 \cdot 0.78908408}{9}$

Step 3.2.2.1.4

Multiply $7$ by $0.78908408$ .

$\sin (C) = \frac{5.52358859}{9}$

Step 3.2.2.1.5

Divide $5.52358859$ by $9$ .

$\sin (C) = 0.61373206$

Step 3.3

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

$C = \arcsin (0.61373206)$

Step 3.4

Simplify the right side.

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Step 3.4.1

Evaluate $\arcsin (0.61373206)$ .

$C = 37.85984761$

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.

$C = 180 - 37.85984761$

Step 3.6

Subtract $37.85984761$ from $180$ .

$C = 142.14015238$

Step 3.7

The solution to the equation $C = 37.85984761$ .

$C = 37.85984761, 142.14015238$

Step 3.8

Exclude the invalid angle.

$C = 37.85984761$

Step 4

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

$A + 37.85984761 + 52.1 = 180$

Step 5

Solve the equation for $A$ .

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Step 5.1

Add $37.85984761$ and $52.1$ .

$A + 89.95984761 = 180$

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 $89.95984761$ from both sides of the equation.

$A = 180 - 89.95984761$

Step 5.2.2

Subtract $89.95984761$ from $180$ .

$A = 90.04015238$

Step 6

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

$a^{2} = b^{2} + c^{2} - 2 b c \cos (A)$

Step 7

Solve the equation.

$a = \sqrt{b^{2} + c^{2} - 2 b c \cos (A)}$

Step 8

Substitute the known values into the equation.

$a = \sqrt{{(54)}^{2} + {(42)}^{2} - 2 \cdot 54 \cdot 42 \cos (90.04015238)}$

Step 9

Simplify the results.

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Step 9.1

Raise $54$ to the power of $2$ .

$a = \sqrt{2916 + {(42)}^{2} - 2 \cdot 54 \cdot (42 \cos (90.04015238))}$

Step 9.2

Raise $42$ to the power of $2$ .

$a = \sqrt{2916 + 1764 - 2 \cdot 54 \cdot (42 \cos (90.04015238))}$

Step 9.3

Multiply $- 2 \cdot 54 \cdot 42$ .

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Step 9.3.1

Multiply $- 2$ by $54$ .

$a = \sqrt{2916 + 1764 - 108 \cdot (42 \cos (90.04015238))}$

Step 9.3.2

Multiply $- 108$ by $42$ .

$a = \sqrt{2916 + 1764 - 4536 \cos (90.04015238)}$

Step 9.4

Add $2916$ and $1764$ .

$a = \sqrt{4680 - 4536 \cos (90.04015238)}$

Step 10

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

$A = 90.04015238$

$B = 52.1$

$C = 37.85984761$

$a = \sqrt{4680 - 4536 \cos (90.04015238)}$

$b = 54$

$c = 42$