Trigonometry Examples

$B = 72.24$ , $c = 16$ , $a = 73$

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.

$b = \sqrt{{(73)}^{2} + {(16)}^{2} - 2 \cdot 73 \cdot 16 \cos (72.24)}$

Step 4

Simplify the results.

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Step 4.1

Raise $73$ to the power of $2$ .

$b = \sqrt{5329 + {(16)}^{2} - 2 \cdot 73 \cdot (16 \cos (72.24))}$

Step 4.2

Raise $16$ to the power of $2$ .

$b = \sqrt{5329 + 256 - 2 \cdot 73 \cdot (16 \cos (72.24))}$

Step 4.3

Multiply $- 2 \cdot 73 \cdot 16$ .

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Step 4.3.1

Multiply $- 2$ by $73$ .

$b = \sqrt{5329 + 256 - 146 \cdot (16 \cos (72.24))}$

Step 4.3.2

Multiply $- 146$ by $16$ .

$b = \sqrt{5329 + 256 - 2336 \cos (72.24)}$

Step 4.4

Add $5329$ and $256$ .

$b = \sqrt{5585 - 2336 \cos (72.24)}$

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 $A$ .

$\frac{\sin (A)}{73} = \frac{\sin (72.24)}{\sqrt{5585 - 2336 \cos (72.24)}}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $73$ .

$73 \frac{\sin (A)}{73} = 73 \frac{\sin (72.24)}{\sqrt{5585 - 2336 \cos (72.24)}}$

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 $73$ .

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Step 7.2.1.1.1

Cancel the common factor.

$73 \frac{\sin (A)}{73} = 73 \frac{\sin (72.24)}{\sqrt{5585 - 2336 \cos (72.24)}}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 73 \frac{\sin (72.24)}{\sqrt{5585 - 2336 \cos (72.24)}}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $73 \frac{\sin (72.24)}{\sqrt{5585 - 2336 \cos (72.24)}}$ .

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Step 7.2.2.1.1

Evaluate $\sin (72.24)$ .

$\sin (A) = 73 \frac{0.95234257}{\sqrt{5585 - 2336 \cos (72.24)}}$

Step 7.2.2.1.2

Multiply $\frac{0.95234257}{\sqrt{5585 - 2336 \cos (72.24)}}$ by $\frac{\sqrt{5585 - 2336 \cos (72.24)}}{\sqrt{5585 - 2336 \cos (72.24)}}$ .

$\sin (A) = 73 (\frac{0.95234257}{\sqrt{5585 - 2336 \cos (72.24)}} \cdot \frac{\sqrt{5585 - 2336 \cos (72.24)}}{\sqrt{5585 - 2336 \cos (72.24)}})$

Step 7.2.2.1.3

Combine and simplify the denominator.

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Step 7.2.2.1.3.1

Multiply $\frac{0.95234257}{\sqrt{5585 - 2336 \cos (72.24)}}$ by $\frac{\sqrt{5585 - 2336 \cos (72.24)}}{\sqrt{5585 - 2336 \cos (72.24)}}$ .

$\sin (A) = 73 \frac{0.95234257 \sqrt{5585 - 2336 \cos (72.24)}}{\sqrt{5585 - 2336 \cos (72.24)} \sqrt{5585 - 2336 \cos (72.24)}}$

Step 7.2.2.1.3.2

Raise $\sqrt{5585 - 2336 \cos (72.24)}$ to the power of $1$ .

$\sin (A) = 73 \frac{0.95234257 \sqrt{5585 - 2336 \cos (72.24)}}{{\sqrt{5585 - 2336 \cos (72.24)}}^{1} \sqrt{5585 - 2336 \cos (72.24)}}$

Step 7.2.2.1.3.3

Raise $\sqrt{5585 - 2336 \cos (72.24)}$ to the power of $1$ .

$\sin (A) = 73 \frac{0.95234257 \sqrt{5585 - 2336 \cos (72.24)}}{{\sqrt{5585 - 2336 \cos (72.24)}}^{1} {\sqrt{5585 - 2336 \cos (72.24)}}^{1}}$

Step 7.2.2.1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sin (A) = 73 \frac{0.95234257 \sqrt{5585 - 2336 \cos (72.24)}}{{\sqrt{5585 - 2336 \cos (72.24)}}^{1 + 1}}$

Step 7.2.2.1.3.5

Add $1$ and $1$ .

$\sin (A) = 73 \frac{0.95234257 \sqrt{5585 - 2336 \cos (72.24)}}{{\sqrt{5585 - 2336 \cos (72.24)}}^{2}}$

Step 7.2.2.1.3.6

Rewrite ${\sqrt{5585 - 2336 \cos (72.24)}}^{2}$ as $5585 - 2336 \cos (72.24)$ .

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Step 7.2.2.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5585 - 2336 \cos (72.24)}$ as ${(5585 - 2336 \cos (72.24))}^{\frac{1}{2}}$ .

$\sin (A) = 73 \frac{0.95234257 \sqrt{5585 - 2336 \cos (72.24)}}{{({(5585 - 2336 \cos (72.24))}^{\frac{1}{2}})}^{2}}$

Step 7.2.2.1.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sin (A) = 73 \frac{0.95234257 \sqrt{5585 - 2336 \cos (72.24)}}{{(5585 - 2336 \cos (72.24))}^{\frac{1}{2} \cdot 2}}$

Step 7.2.2.1.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\sin (A) = 73 \frac{0.95234257 \sqrt{5585 - 2336 \cos (72.24)}}{{(5585 - 2336 \cos (72.24))}^{\frac{2}{2}}}$

Step 7.2.2.1.3.6.4

Cancel the common factor of $2$ .

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Step 7.2.2.1.3.6.4.1

Cancel the common factor.

$\sin (A) = 73 \frac{0.95234257 \sqrt{5585 - 2336 \cos (72.24)}}{{(5585 - 2336 \cos (72.24))}^{\frac{2}{2}}}$

Step 7.2.2.1.3.6.4.2

Rewrite the expression.

$\sin (A) = 73 \frac{0.95234257 \sqrt{5585 - 2336 \cos (72.24)}}{{(5585 - 2336 \cos (72.24))}^{1}}$

Step 7.2.2.1.3.6.5

Simplify.

$\sin (A) = 73 \frac{0.95234257 \sqrt{5585 - 2336 \cos (72.24)}}{5585 - 2336 \cos (72.24)}$

Step 7.2.2.1.4

Multiply $0.95234257$ by $\sqrt{5585 - 2336 \cos (72.24)}$ .

$\sin (A) = 73 \frac{66.47629994}{5585 - 2336 \cos (72.24)}$

Step 7.2.2.1.5

Evaluate $\cos (72.24)$ .

$\sin (A) = 73 \frac{66.47629994}{5585 - 2336 \cdot 0.30503051}$

Step 7.2.2.1.6

Multiply $- 2336$ by $0.30503051$ .

$\sin (A) = 73 \frac{66.47629994}{5585 - 712.55129193}$

Step 7.2.2.1.7

Subtract $712.55129193$ from $5585$ .

$\sin (A) = 73 (\frac{66.47629994}{4872.44870806})$

Step 7.2.2.1.8

Divide $66.47629994$ by $4872.44870806$ .

$\sin (A) = 73 \cdot 0.0136433$

Step 7.2.2.1.9

Multiply $73$ by $0.0136433$ .

$\sin (A) = 0.9959612$

Step 7.3

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

$A = \arcsin (0.9959612)$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (0.9959612)$ .

$A = 84.84878464$

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.

$A = 180 - 84.84878464$

Step 7.6

Subtract $84.84878464$ from $180$ .

$A = 95.15121535$

Step 7.7

The solution to the equation $A = 84.84878464$ .

$A = 84.84878464, 95.15121535$

Step 8

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

$84.84878464 + C + 72.24 = 180$

Step 9

Solve the equation for $C$ .

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Step 9.1

Add $84.84878464$ and $72.24$ .

$C + 157.08878464 = 180$

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 $157.08878464$ from both sides of the equation.

$C = 180 - 157.08878464$

Step 9.2.2

Subtract $157.08878464$ from $180$ .

$C = 22.91121535$

Step 10

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

$A = 84.84878464$

$B = 72.24$

$C = 22.91121535$

$a = 73$

$b = \sqrt{5585 - 2336 \cos (72.24)}$

$c = 16$