Trigonometry Examples

$C = 3$ , $a = 16$ , $b = 33$

Step 1

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

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

Step 2

Solve the equation.

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

Step 3

Substitute the known values into the equation.

$c = \sqrt{{(16)}^{2} + {(33)}^{2} - 2 \cdot 16 \cdot 33 \cos (3)}$

Step 4

Simplify the results.

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Step 4.1

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

$c = \sqrt{256 + {(33)}^{2} - 2 \cdot 16 \cdot (33 \cos (3))}$

Step 4.2

Raise $33$ to the power of $2$ .

$c = \sqrt{256 + 1089 - 2 \cdot 16 \cdot (33 \cos (3))}$

Step 4.3

Multiply $- 2 \cdot 16 \cdot 33$ .

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Step 4.3.1

Multiply $- 2$ by $16$ .

$c = \sqrt{256 + 1089 - 32 \cdot (33 \cos (3))}$

Step 4.3.2

Multiply $- 32$ by $33$ .

$c = \sqrt{256 + 1089 - 1056 \cos (3)}$

Step 4.4

Evaluate $\cos (3)$ .

$c = \sqrt{256 + 1089 - 1056 \cdot 0.99862953}$

Step 4.5

Multiply $- 1056$ by $0.99862953$ .

$c = \sqrt{256 + 1089 - 1054.5527887}$

Step 4.6

Add $256$ and $1089$ .

$c = \sqrt{1345 - 1054.5527887}$

Step 4.7

Subtract $1054.5527887$ from $1345$ .

$c = \sqrt{290.44721129}$

Step 4.8

Evaluate the root.

$c = 17.04251188$

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 $A$ .

$\frac{\sin (A)}{16} = \frac{\sin (3)}{17.04251188}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $16$ .

$16 \frac{\sin (A)}{16} = 16 \frac{\sin (3)}{17.04251188}$

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 $16$ .

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Step 7.2.1.1.1

Cancel the common factor.

$16 \frac{\sin (A)}{16} = 16 \frac{\sin (3)}{17.04251188}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 16 \frac{\sin (3)}{17.04251188}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $16 \frac{\sin (3)}{17.04251188}$ .

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Step 7.2.2.1.1

Evaluate $\sin (3)$ .

$\sin (A) = 16 (\frac{0.05233595}{17.04251188})$

Step 7.2.2.1.2

Divide $0.05233595$ by $17.04251188$ .

$\sin (A) = 16 \cdot 0.0030709$

Step 7.2.2.1.3

Multiply $16$ by $0.0030709$ .

$\sin (A) = 0.04913449$

Step 7.3

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

$A = \arcsin (0.04913449)$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (0.04913449)$ .

$A = 2.81633345$

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.

$A = 180 - 2.81633345$

Step 7.6

Subtract $2.81633345$ from $180$ .

$A = 177.18366654$

Step 7.7

The solution to the equation $A = 2.81633345$ .

$A = 2.81633345, 177.18366654$

Step 7.8

Exclude the invalid angle.

$A = 2.81633345$

Step 8

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

$2.81633345 + 3 + B = 180$

Step 9

Solve the equation for $B$ .

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Step 9.1

Add $2.81633345$ and $3$ .

$5.81633345 + B = 180$

Step 9.2

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

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Step 9.2.1

Subtract $5.81633345$ from both sides of the equation.

$B = 180 - 5.81633345$

Step 9.2.2

Subtract $5.81633345$ from $180$ .

$B = 174.18366654$

Step 10

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

$A = 2.81633345$

$B = 174.18366654$

$C = 3$

$a = 16$

$b = 33$

$c = 17.04251188$