Trigonometry Examples

$A = 60$ , $b = 6$ , $c = 9$

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.

$a = \sqrt{{(6)}^{2} + {(9)}^{2} - 2 \cdot 6 \cdot 9 \cos (60)}$

Step 4

Simplify the results.

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Step 4.1

Raise $6$ to the power of $2$ .

$a = \sqrt{36 + {(9)}^{2} - 2 \cdot 6 \cdot (9 \cos (60))}$

Step 4.2

Raise $9$ to the power of $2$ .

$a = \sqrt{36 + 81 - 2 \cdot 6 \cdot (9 \cos (60))}$

Step 4.3

Multiply $- 2 \cdot 6 \cdot 9$ .

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Step 4.3.1

Multiply $- 2$ by $6$ .

$a = \sqrt{36 + 81 - 12 \cdot (9 \cos (60))}$

Step 4.3.2

Multiply $- 12$ by $9$ .

$a = \sqrt{36 + 81 - 108 \cos (60)}$

Step 4.4

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$a = \sqrt{36 + 81 - 108 (\frac{1}{2})}$

Step 4.5

Cancel the common factor of $2$ .

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Step 4.5.1

Factor $2$ out of $- 108$ .

$a = \sqrt{36 + 81 + 2 (- 54) (\frac{1}{2})}$

Step 4.5.2

Cancel the common factor.

$a = \sqrt{36 + 81 + 2 \cdot (- 54 (\frac{1}{2}))}$

Step 4.5.3

Rewrite the expression.

$a = \sqrt{36 + 81 - 54}$

Step 4.6

Simplify by adding and subtracting.

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Step 4.6.1

Add $36$ and $81$ .

$a = \sqrt{117 - 54}$

Step 4.6.2

Subtract $54$ from $117$ .

$a = \sqrt{63}$

Step 4.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 4.7.1

Factor $9$ out of $63$ .

$a = \sqrt{9 (7)}$

Step 4.7.2

Rewrite $9$ as $3^{2}$ .

$a = \sqrt{3^{2} \cdot 7}$

Step 4.8

Pull terms out from under the radical.

$a = 3 \sqrt{7}$

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 $B$ .

$\frac{\sin (B)}{6} = \frac{\sin (60)}{3 \sqrt{7}}$

Step 7

Solve the equation for $B$ .

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Step 7.1

Multiply both sides of the equation by $6$ .

$6 \frac{\sin (B)}{6} = 6 \frac{\sin (60)}{3 \sqrt{7}}$

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 $6$ .

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Step 7.2.1.1.1

Cancel the common factor.

$6 \frac{\sin (B)}{6} = 6 \frac{\sin (60)}{3 \sqrt{7}}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (B) = 6 \frac{\sin (60)}{3 \sqrt{7}}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $6 \frac{\sin (60)}{3 \sqrt{7}}$ .

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Step 7.2.2.1.1

Cancel the common factor of $3$ .

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Step 7.2.2.1.1.1

Factor $3$ out of $6$ .

$\sin (B) = 3 (2) \frac{\sin (60)}{3 \sqrt{7}}$

Step 7.2.2.1.1.2

Factor $3$ out of $3 \sqrt{7}$ .

$\sin (B) = 3 (2) \frac{\sin (60)}{3 (\sqrt{7})}$

Step 7.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 3 \cdot 2 \frac{\sin (60)}{3 \sqrt{7}}$

Step 7.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 2 \frac{\sin (60)}{\sqrt{7}}$

Step 7.2.2.1.2

Combine $2$ and $\frac{\sin (60)}{\sqrt{7}}$ .

$\sin (B) = \frac{2 \sin (60)}{\sqrt{7}}$

Step 7.2.2.1.3

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$\sin (B) = \frac{2 \frac{\sqrt{3}}{2}}{\sqrt{7}}$

Step 7.2.2.1.4

Combine $2$ and $\frac{\sqrt{3}}{2}$ .

$\sin (B) = \frac{\frac{2 \sqrt{3}}{2}}{\sqrt{7}}$

Step 7.2.2.1.5

Reduce the expression by cancelling the common factors.

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Step 7.2.2.1.5.1

Reduce the expression $\frac{2 \sqrt{3}}{2}$ by cancelling the common factors.

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Step 7.2.2.1.5.1.1

Cancel the common factor.

$\sin (B) = \frac{\frac{2 \sqrt{3}}{2}}{\sqrt{7}}$

Step 7.2.2.1.5.1.2

Rewrite the expression.

$\sin (B) = \frac{\frac{\sqrt{3}}{1}}{\sqrt{7}}$

Step 7.2.2.1.5.2

Divide $\sqrt{3}$ by $1$ .

$\sin (B) = \frac{\sqrt{3}}{\sqrt{7}}$

Step 7.2.2.1.6

Multiply $\frac{\sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sin (B) = \frac{\sqrt{3}}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 7.2.2.1.7

Combine and simplify the denominator.

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Step 7.2.2.1.7.1

Multiply $\frac{\sqrt{3}}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sin (B) = \frac{\sqrt{3} \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 7.2.2.1.7.2

Raise $\sqrt{7}$ to the power of $1$ .

$\sin (B) = \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}}$

Step 7.2.2.1.7.3

Raise $\sqrt{7}$ to the power of $1$ .

$\sin (B) = \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}}$

Step 7.2.2.1.7.4

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

$\sin (B) = \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 7.2.2.1.7.5

Add $1$ and $1$ .

$\sin (B) = \frac{\sqrt{3} \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 7.2.2.1.7.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 7.2.2.1.7.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sin (B) = \frac{\sqrt{3} \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 7.2.2.1.7.6.2

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

$\sin (B) = \frac{\sqrt{3} \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 7.2.2.1.7.6.3

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

$\sin (B) = \frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 7.2.2.1.7.6.4

Cancel the common factor of $2$ .

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Step 7.2.2.1.7.6.4.1

Cancel the common factor.

$\sin (B) = \frac{\sqrt{3} \sqrt{7}}{7^{\frac{2}{2}}}$

Step 7.2.2.1.7.6.4.2

Rewrite the expression.

$\sin (B) = \frac{\sqrt{3} \sqrt{7}}{7^{1}}$

Step 7.2.2.1.7.6.5

Evaluate the exponent.

$\sin (B) = \frac{\sqrt{3} \sqrt{7}}{7}$

Step 7.2.2.1.8

Simplify the numerator.

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Step 7.2.2.1.8.1

Combine using the product rule for radicals.

$\sin (B) = \frac{\sqrt{3 \cdot 7}}{7}$

Step 7.2.2.1.8.2

Multiply $3$ by $7$ .

$\sin (B) = \frac{\sqrt{21}}{7}$

Step 7.3

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

$B = \arcsin (\frac{\sqrt{21}}{7})$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (\frac{\sqrt{21}}{7})$ .

$B = 40.89339464$

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.

$B = 180 - 40.89339464$

Step 7.6

Subtract $40.89339464$ from $180$ .

$B = 139.10660535$

Step 7.7

The solution to the equation $B = 40.89339464$ .

$B = 40.89339464, 139.10660535$

Step 7.8

Exclude the invalid angle.

$B = 40.89339464$

Step 8

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

$60 + C + 40.89339464 = 180$

Step 9

Solve the equation for $C$ .

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Step 9.1

Add $60$ and $40.89339464$ .

$C + 100.89339464 = 180$

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 $100.89339464$ from both sides of the equation.

$C = 180 - 100.89339464$

Step 9.2.2

Subtract $100.89339464$ from $180$ .

$C = 79.10660535$

Step 10

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

$A = 60$

$B = 40.89339464$

$C = 79.10660535$

$a = 3 \sqrt{7}$

$b = 6$

$c = 9$