Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 36 \\ c = \\ a = & 24 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 ° \end{matrix} \end{matrix}$

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.

$c = \sqrt{{(24)}^{2} + {(36)}^{2} - 2 \cdot 24 \cdot 36 \cos (90 °)}$

Step 4

Simplify the results.

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Step 4.1

Raise $24$ to the power of $2$ .

$c = \sqrt{576 + {(36)}^{2} - 2 \cdot 24 \cdot (36 \cos (90 °))}$

Step 4.2

Raise $36$ to the power of $2$ .

$c = \sqrt{576 + 1296 - 2 \cdot 24 \cdot (36 \cos (90 °))}$

Step 4.3

Multiply $- 2 \cdot 24 \cdot 36$ .

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Step 4.3.1

Multiply $- 2$ by $24$ .

$c = \sqrt{576 + 1296 - 48 \cdot (36 \cos (90 °))}$

Step 4.3.2

Multiply $- 48$ by $36$ .

$c = \sqrt{576 + 1296 - 1728 \cos (90 °)}$

Step 4.4

The exact value of $\cos (90 °)$ is $0$ .

$c = \sqrt{576 + 1296 - 1728 \cdot 0}$

Step 4.5

Multiply $- 1728$ by $0$ .

$c = \sqrt{576 + 1296 + 0}$

Step 4.6

Add $576 + 1296$ and $0$ .

$c = \sqrt{576 + 1296}$

Step 4.7

Add $576$ and $1296$ .

$c = \sqrt{1872}$

Step 4.8

Rewrite $1872$ as $12^{2} \cdot 13$ .

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Step 4.8.1

Factor $144$ out of $1872$ .

$c = \sqrt{144 (13)}$

Step 4.8.2

Rewrite $144$ as $12^{2}$ .

$c = \sqrt{12^{2} \cdot 13}$

Step 4.9

Pull terms out from under the radical.

$c = 12 \sqrt{13}$

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 $A$ .

$\frac{\sin (A)}{24} = \frac{\sin (90 °)}{12 \sqrt{13}}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $24$ .

$24 \frac{\sin (A)}{24} = 24 \frac{\sin (90 °)}{12 \sqrt{13}}$

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 $24$ .

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Step 7.2.1.1.1

Cancel the common factor.

$24 \frac{\sin (A)}{24} = 24 \frac{\sin (90 °)}{12 \sqrt{13}}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 24 \frac{\sin (90 °)}{12 \sqrt{13}}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $24 \frac{\sin (90 °)}{12 \sqrt{13}}$ .

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Step 7.2.2.1.1

Cancel the common factor of $12$ .

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Step 7.2.2.1.1.1

Factor $12$ out of $24$ .

$\sin (A) = 12 (2) \frac{\sin (90 °)}{12 \sqrt{13}}$

Step 7.2.2.1.1.2

Factor $12$ out of $12 \sqrt{13}$ .

$\sin (A) = 12 (2) \frac{\sin (90 °)}{12 (\sqrt{13})}$

Step 7.2.2.1.1.3

Cancel the common factor.

$\sin (A) = 12 \cdot 2 \frac{\sin (90 °)}{12 \sqrt{13}}$

Step 7.2.2.1.1.4

Rewrite the expression.

$\sin (A) = 2 \frac{\sin (90 °)}{\sqrt{13}}$

Step 7.2.2.1.2

Combine $2$ and $\frac{\sin (90 °)}{\sqrt{13}}$ .

$\sin (A) = \frac{2 \sin (90 °)}{\sqrt{13}}$

Step 7.2.2.1.3

The exact value of $\sin (90 °)$ is $1$ .

$\sin (A) = \frac{2 \cdot 1}{\sqrt{13}}$

Step 7.2.2.1.4

Multiply $2$ by $1$ .

$\sin (A) = \frac{2}{\sqrt{13}}$

Step 7.2.2.1.5

Multiply $\frac{2}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sin (A) = \frac{2}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

Step 7.2.2.1.6

Combine and simplify the denominator.

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Step 7.2.2.1.6.1

Multiply $\frac{2}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$ .

$\sin (A) = \frac{2 \sqrt{13}}{\sqrt{13} \sqrt{13}}$

Step 7.2.2.1.6.2

Raise $\sqrt{13}$ to the power of $1$ .

$\sin (A) = \frac{2 \sqrt{13}}{{\sqrt{13}}^{1} \sqrt{13}}$

Step 7.2.2.1.6.3

Raise $\sqrt{13}$ to the power of $1$ .

$\sin (A) = \frac{2 \sqrt{13}}{{\sqrt{13}}^{1} {\sqrt{13}}^{1}}$

Step 7.2.2.1.6.4

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

$\sin (A) = \frac{2 \sqrt{13}}{{\sqrt{13}}^{1 + 1}}$

Step 7.2.2.1.6.5

Add $1$ and $1$ .

$\sin (A) = \frac{2 \sqrt{13}}{{\sqrt{13}}^{2}}$

Step 7.2.2.1.6.6

Rewrite ${\sqrt{13}}^{2}$ as $13$ .

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Step 7.2.2.1.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{13}$ as $13^{\frac{1}{2}}$ .

$\sin (A) = \frac{2 \sqrt{13}}{{(13^{\frac{1}{2}})}^{2}}$

Step 7.2.2.1.6.6.2

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

$\sin (A) = \frac{2 \sqrt{13}}{13^{\frac{1}{2} \cdot 2}}$

Step 7.2.2.1.6.6.3

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

$\sin (A) = \frac{2 \sqrt{13}}{13^{\frac{2}{2}}}$

Step 7.2.2.1.6.6.4

Cancel the common factor of $2$ .

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Step 7.2.2.1.6.6.4.1

Cancel the common factor.

$\sin (A) = \frac{2 \sqrt{13}}{13^{\frac{2}{2}}}$

Step 7.2.2.1.6.6.4.2

Rewrite the expression.

$\sin (A) = \frac{2 \sqrt{13}}{13^{1}}$

Step 7.2.2.1.6.6.5

Evaluate the exponent.

$\sin (A) = \frac{2 \sqrt{13}}{13}$

Step 7.3

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

$A = \arcsin (\frac{2 \sqrt{13}}{13})$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (\frac{2 \sqrt{13}}{13})$ .

$A = 33.69006752$

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.

$A = 180 - 33.69006752$

Step 7.6

Subtract $33.69006752$ from $180$ .

$A = 146.30993247$

Step 7.7

The solution to the equation $A = 33.69006752$ .

$A = 33.69006752, 146.30993247$

Step 7.8

Exclude the invalid angle.

$A = 33.69006752$

Step 8

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

$33.69006752 + 90 ° + B = 180$

Step 9

Solve the equation for $B$ .

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Step 9.1

Add $33.69006752$ and $90 °$ .

$123.69006752 + B = 180$

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 $123.69006752$ from both sides of the equation.

$B = 180 - 123.69006752$

Step 9.2.2

Subtract $123.69006752$ from $180$ .

$B = 56.30993247$

Step 10

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

$A = 33.69006752$

$B = 56.30993247$

$C = 90 °$

$a = 24$

$b = 36$

$c = 12 \sqrt{13}$