Trigonometry Examples

$B = 72.66$ , $a = 23.78$ , $c = 25.74$

Step 1

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

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

Step 2

Solve the equation.

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

Step 3

Substitute the known values into the equation.

$b = \sqrt{{(23.78)}^{2} + {(25.74)}^{2} - 2 \cdot 23.78 \cdot 25.74 \cos (72.66)}$

Step 4

Simplify the results.

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Step 4.1

Raise $23.78$ to the power of $2$ .

$b = \sqrt{565.4884 + {(25.74)}^{2} - 2 \cdot 23.78 \cdot (25.74 \cos (72.66))}$

Step 4.2

Raise $25.74$ to the power of $2$ .

$b = \sqrt{565.4884 + 662.5476 - 2 \cdot 23.78 \cdot (25.74 \cos (72.66))}$

Step 4.3

Multiply $- 2 \cdot 23.78 \cdot 25.74$ .

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Step 4.3.1

Multiply $- 2$ by $23.78$ .

$b = \sqrt{565.4884 + 662.5476 - 47.56 \cdot (25.74 \cos (72.66))}$

Step 4.3.2

Multiply $- 47.56$ by $25.74$ .

$b = \sqrt{565.4884 + 662.5476 - 1224.1944 \cos (72.66)}$

Step 4.4

Multiply $- 1224.1944$ by $\cos (72.66)$ .

$b = \sqrt{565.4884 + 662.5476 - 364.86055204}$

Step 4.5

Simplify by adding and subtracting.

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Step 4.5.1

Add $565.4884$ and $662.5476$ .

$b = \sqrt{1228.036 - 364.86055204}$

Step 4.5.2

Subtract $364.86055204$ from $1228.036$ .

$b = \sqrt{863.17544795}$

Step 4.6

Evaluate the root.

$b = 29.37984765$

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 $A$ .

$\frac{\sin (A)}{23.78} = \frac{\sin (72.66)}{29.37984765}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $23.78$ .

$23.78 \frac{\sin (A)}{23.78} = 23.78 \frac{\sin (72.66)}{29.37984765}$

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 $23.78$ .

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Step 7.2.1.1.1

Cancel the common factor.

$23.78 \frac{\sin (A)}{23.78} = 23.78 \frac{\sin (72.66)}{29.37984765}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 23.78 \frac{\sin (72.66)}{29.37984765}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $23.78 \frac{\sin (72.66)}{29.37984765}$ .

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Step 7.2.2.1.1

Evaluate $\sin (72.66)$ .

$\sin (A) = 23.78 (\frac{0.95455296}{29.37984765})$

Step 7.2.2.1.2

Divide $0.95455296$ by $29.37984765$ .

$\sin (A) = 23.78 \cdot 0.03249005$

Step 7.2.2.1.3

Multiply $23.78$ by $0.03249005$ .

$\sin (A) = 0.77261358$

Step 7.3

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

$A = \arcsin (0.77261358)$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (0.77261358)$ .

$A = 50.58917004$

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.

$A = 180 - 50.58917004$

Step 7.6

Subtract $50.58917004$ from $180$ .

$A = 129.41082995$

Step 7.7

The solution to the equation $A = 50.58917004$ .

$A = 50.58917004, 129.41082995$

Step 7.8

Exclude the invalid angle.

$A = 50.58917004$

Step 8

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

$50.58917004 + C + 72.66 = 180$

Step 9

Solve the equation for $C$ .

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Step 9.1

Add $50.58917004$ and $72.66$ .

$C + 123.24917004 = 180$

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 $123.24917004$ from both sides of the equation.

$C = 180 - 123.24917004$

Step 9.2.2

Subtract $123.24917004$ from $180$ .

$C = 56.75082995$

Step 10

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

$A = 50.58917004$

$B = 72.66$

$C = 56.75082995$

$a = 23.78$

$b = 29.37984765$

$c = 25.74$