Trigonometry Examples

$C = 56.4$ , $c = 31.3$ , $b = 25.7$

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 $B$ .

$\frac{\sin (B)}{25.7} = \frac{\sin (56.4)}{31.3}$

Step 3

Solve the equation for $B$ .

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Step 3.1

Multiply both sides of the equation by $25.7$ .

$25.7 \frac{\sin (B)}{25.7} = 25.7 \frac{\sin (56.4)}{31.3}$

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 $25.7$ .

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Step 3.2.1.1.1

Cancel the common factor.

$25.7 \frac{\sin (B)}{25.7} = 25.7 \frac{\sin (56.4)}{31.3}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (B) = 25.7 \frac{\sin (56.4)}{31.3}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $25.7 \frac{\sin (56.4)}{31.3}$ .

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Step 3.2.2.1.1

Evaluate $\sin (56.4)$ .

$\sin (B) = 25.7 (\frac{0.83292124}{31.3})$

Step 3.2.2.1.2

Divide $0.83292124$ by $31.3$ .

$\sin (B) = 25.7 \cdot 0.0266109$

Step 3.2.2.1.3

Multiply $25.7$ by $0.0266109$ .

$\sin (B) = 0.68390018$

Step 3.3

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

$B = \arcsin (0.68390018)$

Step 3.4

Simplify the right side.

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Step 3.4.1

Evaluate $\arcsin (0.68390018)$ .

$B = 43.14917444$

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.

$B = 180 - 43.14917444$

Step 3.6

Subtract $43.14917444$ from $180$ .

$B = 136.85082555$

Step 3.7

The solution to the equation $B = 43.14917444$ .

$B = 43.14917444, 136.85082555$

Step 3.8

Exclude the invalid angle.

$B = 43.14917444$

Step 4

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

$A + 56.4 + 43.14917444 = 180$

Step 5

Solve the equation for $A$ .

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Step 5.1

Add $56.4$ and $43.14917444$ .

$A + 99.54917444 = 180$

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 $99.54917444$ from both sides of the equation.

$A = 180 - 99.54917444$

Step 5.2.2

Subtract $99.54917444$ from $180$ .

$A = 80.45082555$

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.

$a = \sqrt{{(25.7)}^{2} + {(31.3)}^{2} - 2 \cdot 25.7 \cdot 31.3 \cos (80.45082555)}$

Step 9

Simplify the results.

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Step 9.1

Raise $25.7$ to the power of $2$ .

$a = \sqrt{660.49 + {(31.3)}^{2} - 2 \cdot 25.7 \cdot (31.3 \cos (80.45082555))}$

Step 9.2

Raise $31.3$ to the power of $2$ .

$a = \sqrt{660.49 + 979.69 - 2 \cdot 25.7 \cdot (31.3 \cos (80.45082555))}$

Step 9.3

Multiply $- 2 \cdot 25.7 \cdot 31.3$ .

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Step 9.3.1

Multiply $- 2$ by $25.7$ .

$a = \sqrt{660.49 + 979.69 - 51.4 \cdot (31.3 \cos (80.45082555))}$

Step 9.3.2

Multiply $- 51.4$ by $31.3$ .

$a = \sqrt{660.49 + 979.69 - 1608.82 \cos (80.45082555)}$

Step 9.4

Multiply $- 1608.82$ by $\cos (80.45082555)$ .

$a = \sqrt{660.49 + 979.69 - 266.89363405}$

Step 9.5

Simplify by adding and subtracting.

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Step 9.5.1

Add $660.49$ and $979.69$ .

$a = \sqrt{1640.18 - 266.89363405}$

Step 9.5.2

Subtract $266.89363405$ from $1640.18$ .

$a = \sqrt{1373.28636594}$

Step 9.6

Evaluate the root.

$a = 37.05787859$

Step 10

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

$A = 80.45082555$

$B = 43.14917444$

$C = 56.4$

$a = 37.05787859$

$b = 25.7$

$c = 31.3$