Trigonometry Examples

$A = 51 °$ , $b = 11$ , $c = 7$

Step 1

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 2

Solve the equation.

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

Step 3

Substitute the known values into the equation.

$a = \sqrt{{(11)}^{2} + {(7)}^{2} - 2 \cdot 11 \cdot 7 \cos (51 °)}$

Step 4

Simplify the results.

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Step 4.1

Raise $11$ to the power of $2$ .

$a = \sqrt{121 + {(7)}^{2} - 2 \cdot 11 \cdot (7 \cos (51 °))}$

Step 4.2

Raise $7$ to the power of $2$ .

$a = \sqrt{121 + 49 - 2 \cdot 11 \cdot (7 \cos (51 °))}$

Step 4.3

Multiply $- 2 \cdot 11 \cdot 7$ .

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Step 4.3.1

Multiply $- 2$ by $11$ .

$a = \sqrt{121 + 49 - 22 \cdot (7 \cos (51 °))}$

Step 4.3.2

Multiply $- 22$ by $7$ .

$a = \sqrt{121 + 49 - 154 \cos (51 °)}$

Step 4.4

Evaluate $\cos (51 °)$ .

$a = \sqrt{121 + 49 - 154 \cdot 0.62932039}$

Step 4.5

Multiply $- 154$ by $0.62932039$ .

$a = \sqrt{121 + 49 - 96.91534022}$

Step 4.6

Add $121$ and $49$ .

$a = \sqrt{170 - 96.91534022}$

Step 4.7

Subtract $96.91534022$ from $170$ .

$a = \sqrt{73.08465977}$

Step 4.8

Evaluate the root.

$a = 8.54895664$

Step 5

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 6

Substitute the known values into the law of sines to find $B$ .

$\frac{\sin (B)}{11} = \frac{\sin (51 °)}{8.54895664}$

Step 7

Solve the equation for $B$ .

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Step 7.1

Multiply both sides of the equation by $11$ .

$11 \frac{\sin (B)}{11} = 11 \frac{\sin (51 °)}{8.54895664}$

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $11$ .

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Step 7.2.1.1.1

Cancel the common factor.

$11 \frac{\sin (B)}{11} = 11 \frac{\sin (51 °)}{8.54895664}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (B) = 11 \frac{\sin (51 °)}{8.54895664}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $11 \frac{\sin (51 °)}{8.54895664}$ .

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Step 7.2.2.1.1

Evaluate $\sin (51 °)$ .

$\sin (B) = 11 (\frac{0.77714596}{8.54895664})$

Step 7.2.2.1.2

Divide $0.77714596$ by $8.54895664$ .

$\sin (B) = 11 \cdot 0.09090535$

Step 7.2.2.1.3

Multiply $11$ by $0.09090535$ .

$\sin (B) = 0.99995893$

Step 7.3

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

$B = \arcsin (0.99995893)$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (0.99995893)$ .

$B = 89.48074467$

Step 7.5

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.

$B = 180 - 89.48074467$

Step 7.6

Subtract $89.48074467$ from $180$ .

$B = 90.51925532$

Step 7.7

The solution to the equation $B = 89.48074467$ .

$B = 89.48074467, 90.51925532$

Step 8

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

$51 ° + C + 89.48074467 = 180$

Step 9

Solve the equation for $C$ .

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Step 9.1

Add $51 °$ and $89.48074467$ .

$C + 140.48074467 = 180$

Step 9.2

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

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Step 9.2.1

Subtract $140.48074467$ from both sides of the equation.

$C = 180 - 140.48074467$

Step 9.2.2

Subtract $140.48074467$ from $180$ .

$C = 39.51925532$

Step 10

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

$A = 51 °$

$B = 89.48074467$

$C = 39.51925532$

$a = 8.54895664$

$b = 11$

$c = 7$