Trigonometry Examples

$B = 110$ , $b = 5$ , $c = 4$

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 $C$ .

$\frac{\sin (C)}{4} = \frac{\sin (110)}{5}$

Step 3

Solve the equation for $C$ .

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Step 3.1

Multiply both sides of the equation by $4$ .

$4 \frac{\sin (C)}{4} = 4 \frac{\sin (110)}{5}$

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 $4$ .

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Step 3.2.1.1.1

Cancel the common factor.

$4 \frac{\sin (C)}{4} = 4 \frac{\sin (110)}{5}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (C) = 4 \frac{\sin (110)}{5}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $4 \frac{\sin (110)}{5}$ .

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Step 3.2.2.1.1

Evaluate $\sin (110)$ .

$\sin (C) = 4 (\frac{0.93969262}{5})$

Step 3.2.2.1.2

Divide $0.93969262$ by $5$ .

$\sin (C) = 4 \cdot 0.18793852$

Step 3.2.2.1.3

Multiply $4$ by $0.18793852$ .

$\sin (C) = 0.75175409$

Step 3.3

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

$C = \arcsin (0.75175409)$

Step 3.4

Simplify the right side.

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Step 3.4.1

Evaluate $\arcsin (0.75175409)$ .

$C = 48.74255245$

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.

$C = 180 - 48.74255245$

Step 3.6

Subtract $48.74255245$ from $180$ .

$C = 131.25744754$

Step 3.7

The solution to the equation $C = 48.74255245$ .

$C = 48.74255245, 131.25744754$

Step 3.8

Exclude the invalid angle.

$C = 48.74255245$

Step 4

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

$A + 48.74255245 + 110 = 180$

Step 5

Solve the equation for $A$ .

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Step 5.1

Add $48.74255245$ and $110$ .

$A + 158.74255245 = 180$

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 $158.74255245$ from both sides of the equation.

$A = 180 - 158.74255245$

Step 5.2.2

Subtract $158.74255245$ from $180$ .

$A = 21.25744754$

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.

$a = \sqrt{{(5)}^{2} + {(4)}^{2} - 2 \cdot 5 \cdot 4 \cos (21.25744754)}$

Step 9

Simplify the results.

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Step 9.1

Raise $5$ to the power of $2$ .

$a = \sqrt{25 + {(4)}^{2} - 2 \cdot 5 \cdot (4 \cos (21.25744754))}$

Step 9.2

Raise $4$ to the power of $2$ .

$a = \sqrt{25 + 16 - 2 \cdot 5 \cdot (4 \cos (21.25744754))}$

Step 9.3

Multiply $- 2 \cdot 5 \cdot 4$ .

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Step 9.3.1

Multiply $- 2$ by $5$ .

$a = \sqrt{25 + 16 - 10 \cdot (4 \cos (21.25744754))}$

Step 9.3.2

Multiply $- 10$ by $4$ .

$a = \sqrt{25 + 16 - 40 \cos (21.25744754)}$

Step 9.4

Add $25$ and $16$ .

$a = \sqrt{41 - 40 \cos (21.25744754)}$

Step 10

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

$A = 21.25744754$

$B = 110$

$C = 48.74255245$

$a = \sqrt{41 - 40 \cos (21.25744754)}$

$b = 5$

$c = 4$