Trigonometry Examples

$a = 3$ , $b = 4$ , $C = 30$

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.

$c = \sqrt{{(3)}^{2} + {(4)}^{2} - 2 \cdot 3 \cdot 4 \cos (30)}$

Step 4

Simplify the results.

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Step 4.1

Raise $3$ to the power of $2$ .

$c = \sqrt{9 + {(4)}^{2} - 2 \cdot 3 \cdot (4 \cos (30))}$

Step 4.2

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

$c = \sqrt{9 + 16 - 2 \cdot 3 \cdot (4 \cos (30))}$

Step 4.3

Multiply $- 2 \cdot 3 \cdot 4$ .

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Step 4.3.1

Multiply $- 2$ by $3$ .

$c = \sqrt{9 + 16 - 6 \cdot (4 \cos (30))}$

Step 4.3.2

Multiply $- 6$ by $4$ .

$c = \sqrt{9 + 16 - 24 \cos (30)}$

Step 4.4

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$c = \sqrt{9 + 16 - 24 \frac{\sqrt{3}}{2}}$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Factor $2$ out of $- 24$ .

$c = \sqrt{9 + 16 + 2 (- 12) (\frac{\sqrt{3}}{2})}$

Step 4.5.2

Cancel the common factor.

$c = \sqrt{9 + 16 + 2 \cdot (- 12 \frac{\sqrt{3}}{2})}$

Step 4.5.3

Rewrite the expression.

$c = \sqrt{9 + 16 - 12 \sqrt{3}}$

Step 4.6

Add $9$ and $16$ .

$c = \sqrt{25 - 12 \sqrt{3}}$

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 $A$ .

$\frac{\sin (A)}{3} = \frac{\sin (30)}{\sqrt{25 - 12 \sqrt{3}}}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $3$ .

$3 \frac{\sin (A)}{3} = 3 \frac{\sin (30)}{\sqrt{25 - 12 \sqrt{3}}}$

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 $3$ .

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Step 7.2.1.1.1

Cancel the common factor.

$3 \frac{\sin (A)}{3} = 3 \frac{\sin (30)}{\sqrt{25 - 12 \sqrt{3}}}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 3 \frac{\sin (30)}{\sqrt{25 - 12 \sqrt{3}}}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $3 \frac{\sin (30)}{\sqrt{25 - 12 \sqrt{3}}}$ .

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Step 7.2.2.1.1

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$\sin (A) = 3 \frac{\frac{1}{2}}{\sqrt{25 - 12 \sqrt{3}}}$

Step 7.2.2.1.2

Multiply the numerator by the reciprocal of the denominator.

$\sin (A) = 3 (\frac{1}{2} \cdot \frac{1}{\sqrt{25 - 12 \sqrt{3}}})$

Step 7.2.2.1.3

Multiply $\frac{1}{2}$ by $\frac{1}{\sqrt{25 - 12 \sqrt{3}}}$ .

$\sin (A) = 3 \frac{1}{2 \sqrt{25 - 12 \sqrt{3}}}$

Step 7.2.2.1.4

Multiply $\frac{1}{2 \sqrt{25 - 12 \sqrt{3}}}$ by $\frac{\sqrt{25 - 12 \sqrt{3}}}{\sqrt{25 - 12 \sqrt{3}}}$ .

$\sin (A) = 3 (\frac{1}{2 \sqrt{25 - 12 \sqrt{3}}} \cdot \frac{\sqrt{25 - 12 \sqrt{3}}}{\sqrt{25 - 12 \sqrt{3}}})$

Step 7.2.2.1.5

Combine and simplify the denominator.

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Step 7.2.2.1.5.1

Multiply $\frac{1}{2 \sqrt{25 - 12 \sqrt{3}}}$ by $\frac{\sqrt{25 - 12 \sqrt{3}}}{\sqrt{25 - 12 \sqrt{3}}}$ .

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 \sqrt{25 - 12 \sqrt{3}} \sqrt{25 - 12 \sqrt{3}}}$

Step 7.2.2.1.5.2

Move $\sqrt{25 - 12 \sqrt{3}}$ .

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 (\sqrt{25 - 12 \sqrt{3}} \sqrt{25 - 12 \sqrt{3}})}$

Step 7.2.2.1.5.3

Raise $\sqrt{25 - 12 \sqrt{3}}$ to the power of $1$ .

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 ({\sqrt{25 - 12 \sqrt{3}}}^{1} \sqrt{25 - 12 \sqrt{3}})}$

Step 7.2.2.1.5.4

Raise $\sqrt{25 - 12 \sqrt{3}}$ to the power of $1$ .

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 ({\sqrt{25 - 12 \sqrt{3}}}^{1} {\sqrt{25 - 12 \sqrt{3}}}^{1})}$

Step 7.2.2.1.5.5

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

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 {\sqrt{25 - 12 \sqrt{3}}}^{1 + 1}}$

Step 7.2.2.1.5.6

Add $1$ and $1$ .

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 {\sqrt{25 - 12 \sqrt{3}}}^{2}}$

Step 7.2.2.1.5.7

Rewrite ${\sqrt{25 - 12 \sqrt{3}}}^{2}$ as $25 - 12 \sqrt{3}$ .

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Step 7.2.2.1.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{25 - 12 \sqrt{3}}$ as ${(25 - 12 \sqrt{3})}^{\frac{1}{2}}$ .

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 {({(25 - 12 \sqrt{3})}^{\frac{1}{2}})}^{2}}$

Step 7.2.2.1.5.7.2

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

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 {(25 - 12 \sqrt{3})}^{\frac{1}{2} \cdot 2}}$

Step 7.2.2.1.5.7.3

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

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 {(25 - 12 \sqrt{3})}^{\frac{2}{2}}}$

Step 7.2.2.1.5.7.4

Cancel the common factor of $2$ .

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Step 7.2.2.1.5.7.4.1

Cancel the common factor.

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 {(25 - 12 \sqrt{3})}^{\frac{2}{2}}}$

Step 7.2.2.1.5.7.4.2

Rewrite the expression.

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 {(25 - 12 \sqrt{3})}^{1}}$

Step 7.2.2.1.5.7.5

Simplify.

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}}}{2 (25 - 12 \sqrt{3})}$

Step 7.2.2.1.6

Multiply $\frac{\sqrt{25 - 12 \sqrt{3}}}{2 (25 - 12 \sqrt{3})}$ by $\frac{25 + 12 \sqrt{3}}{25 + 12 \sqrt{3}}$ .

$\sin (A) = 3 (\frac{\sqrt{25 - 12 \sqrt{3}}}{2 (25 - 12 \sqrt{3})} \cdot \frac{25 + 12 \sqrt{3}}{25 + 12 \sqrt{3}})$

Step 7.2.2.1.7

Multiply $\frac{\sqrt{25 - 12 \sqrt{3}}}{2 (25 - 12 \sqrt{3})}$ by $\frac{25 + 12 \sqrt{3}}{25 + 12 \sqrt{3}}$ .

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}} (25 + 12 \sqrt{3})}{2 (25 - 12 \sqrt{3}) (25 + 12 \sqrt{3})}$

Step 7.2.2.1.8

Simplify the expression.

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Step 7.2.2.1.8.1

Move $25 - 12 \sqrt{3}$ .

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}} (25 + 12 \sqrt{3})}{2 ((25 - 12 \sqrt{3}) (25 + 12 \sqrt{3}))}$

Step 7.2.2.1.8.2

Expand the denominator using the FOIL method.

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}} (25 + 12 \sqrt{3})}{2 (625 + 300 \sqrt{3} - 300 \sqrt{3} - 144 {\sqrt{3}}^{2})}$

Step 7.2.2.1.8.3

Simplify.

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}} (25 + 12 \sqrt{3})}{2 \cdot 193}$

Step 7.2.2.1.8.4

Multiply $2$ by $193$ .

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}} (25 + 12 \sqrt{3})}{386}$

Step 7.2.2.1.9

Apply the distributive property.

$\sin (A) = 3 \frac{\sqrt{25 - 12 \sqrt{3}} \cdot 25 + \sqrt{25 - 12 \sqrt{3}} (12 \sqrt{3})}{386}$

Step 7.2.2.1.10

Move $25$ to the left of $\sqrt{25 - 12 \sqrt{3}}$ .

$\sin (A) = 3 \frac{25 \cdot \sqrt{25 - 12 \sqrt{3}} + \sqrt{25 - 12 \sqrt{3}} (12 \sqrt{3})}{386}$

Step 7.2.2.1.11

Combine using the product rule for radicals.

$\sin (A) = 3 \frac{25 \sqrt{25 - 12 \sqrt{3}} + 12 \sqrt{(25 - 12 \sqrt{3}) \cdot 3}}{386}$

Step 7.2.2.1.12

Combine $3$ and $\frac{25 \sqrt{25 - 12 \sqrt{3}} + 12 \sqrt{(25 - 12 \sqrt{3}) \cdot 3}}{386}$ .

$\sin (A) = \frac{3 (25 \sqrt{25 - 12 \sqrt{3}} + 12 \sqrt{(25 - 12 \sqrt{3}) \cdot 3})}{386}$

Step 7.2.2.1.13

Move $3$ to the left of $25 - 12 \sqrt{3}$ .

$\sin (A) = \frac{3 (25 \sqrt{25 - 12 \sqrt{3}} + 12 \sqrt{3 (25 - 12 \sqrt{3})})}{386}$

Step 7.3

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

$A = \arcsin (\frac{3 (25 \sqrt{25 - 12 \sqrt{3}} + 12 \sqrt{3 (25 - 12 \sqrt{3})})}{386})$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (\frac{3 (25 \sqrt{25 - 12 \sqrt{3}} + 12 \sqrt{3 (25 - 12 \sqrt{3})})}{386})$ .

$A = 46.93568657$

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.

$A = 180 - 46.93568657$

Step 7.6

Subtract $46.93568657$ from $180$ .

$A = 133.06431342$

Step 7.7

The solution to the equation $A = 46.93568657$ .

$A = 46.93568657, 133.06431342$

Step 8

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

$46.93568657 + 30 + B = 180$

Step 9

Solve the equation for $B$ .

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Step 9.1

Add $46.93568657$ and $30$ .

$76.93568657 + B = 180$

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 $76.93568657$ from both sides of the equation.

$B = 180 - 76.93568657$

Step 9.2.2

Subtract $76.93568657$ from $180$ .

$B = 103.06431342$

Step 10

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

$A = 46.93568657$

$B = 103.06431342$

$C = 30$

$a = 3$

$b = 4$

$c = \sqrt{25 - 12 \sqrt{3}}$