Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 23 \\ c = \\ a = & 34 ° \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 ° \end{matrix} \end{matrix}$

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.

$c = \sqrt{{(34 °)}^{2} + {(23)}^{2} - 2 \cdot 34 ° \cdot 23 \cos (90 °)}$

Step 4

Simplify the results.

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Step 4.1

Raise $34 °$ to the power of $2$ .

$c = \sqrt{1156 + {(23)}^{2} - 2 \cdot 34 ° \cdot (23 \cos (90 °))}$

Step 4.2

Raise $23$ to the power of $2$ .

$c = \sqrt{1156 + 529 - 2 \cdot 34 ° \cdot (23 \cos (90 °))}$

Step 4.3

Multiply $- 2 \cdot 34 ° \cdot 23$ .

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Step 4.3.1

Multiply $- 2$ by $34 °$ .

$c = \sqrt{1156 + 529 - 68 \cdot (23 \cos (90 °))}$

Step 4.3.2

Multiply $- 68$ by $23$ .

$c = \sqrt{1156 + 529 - 1564 \cos (90 °)}$

Step 4.4

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

$c = \sqrt{1156 + 529 - 1564 \cdot 0}$

Step 4.5

Multiply $- 1564$ by $0$ .

$c = \sqrt{1156 + 529 + 0}$

Step 4.6

Add $1156 + 529$ and $0$ .

$c = \sqrt{1156 + 529}$

Step 4.7

Add $1156$ and $529$ .

$c = \sqrt{1685}$

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 $A$ .

$\frac{\sin (A)}{34 °} = \frac{\sin (90 °)}{\sqrt{1685}}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $34 °$ .

$34 ° \frac{\sin (A)}{34 °} = 34 ° \frac{\sin (90 °)}{\sqrt{1685}}$

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 $34 °$ .

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Step 7.2.1.1.1

Cancel the common factor.

$34 ° \frac{\sin (A)}{34 °} = 34 ° \frac{\sin (90 °)}{\sqrt{1685}}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 34 ° \frac{\sin (90 °)}{\sqrt{1685}}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $34 ° \frac{\sin (90 °)}{\sqrt{1685}}$ .

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Step 7.2.2.1.1

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

$\sin (A) = 34 ° \frac{1}{\sqrt{1685}}$

Step 7.2.2.1.2

Multiply $\frac{1}{\sqrt{1685}}$ by $\frac{\sqrt{1685}}{\sqrt{1685}}$ .

$\sin (A) = 34 ° (\frac{1}{\sqrt{1685}} \cdot \frac{\sqrt{1685}}{\sqrt{1685}})$

Step 7.2.2.1.3

Combine and simplify the denominator.

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Step 7.2.2.1.3.1

Multiply $\frac{1}{\sqrt{1685}}$ by $\frac{\sqrt{1685}}{\sqrt{1685}}$ .

$\sin (A) = 34 ° \frac{\sqrt{1685}}{\sqrt{1685} \sqrt{1685}}$

Step 7.2.2.1.3.2

Raise $\sqrt{1685}$ to the power of $1$ .

$\sin (A) = 34 ° \frac{\sqrt{1685}}{{\sqrt{1685}}^{1} \sqrt{1685}}$

Step 7.2.2.1.3.3

Raise $\sqrt{1685}$ to the power of $1$ .

$\sin (A) = 34 ° \frac{\sqrt{1685}}{{\sqrt{1685}}^{1} {\sqrt{1685}}^{1}}$

Step 7.2.2.1.3.4

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

$\sin (A) = 34 ° \frac{\sqrt{1685}}{{\sqrt{1685}}^{1 + 1}}$

Step 7.2.2.1.3.5

Add $1$ and $1$ .

$\sin (A) = 34 ° \frac{\sqrt{1685}}{{\sqrt{1685}}^{2}}$

Step 7.2.2.1.3.6

Rewrite ${\sqrt{1685}}^{2}$ as $1685$ .

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Step 7.2.2.1.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1685}$ as $1685^{\frac{1}{2}}$ .

$\sin (A) = 34 ° \frac{\sqrt{1685}}{{(1685^{\frac{1}{2}})}^{2}}$

Step 7.2.2.1.3.6.2

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

$\sin (A) = 34 ° \frac{\sqrt{1685}}{1685^{\frac{1}{2} \cdot 2}}$

Step 7.2.2.1.3.6.3

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

$\sin (A) = 34 ° \frac{\sqrt{1685}}{1685^{\frac{2}{2}}}$

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.

$\sin (A) = 34 ° \frac{\sqrt{1685}}{1685^{\frac{2}{2}}}$

Step 7.2.2.1.3.6.4.2

Rewrite the expression.

$\sin (A) = 34 ° \frac{\sqrt{1685}}{1685^{1}}$

Step 7.2.2.1.3.6.5

Evaluate the exponent.

$\sin (A) = 34 ° \frac{\sqrt{1685}}{1685}$

Step 7.2.2.1.4

Combine $34 °$ and $\frac{\sqrt{1685}}{1685}$ .

$\sin (A) = \frac{34 ° \sqrt{1685}}{1685}$

Step 7.3

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

$A = \arcsin (\frac{34 ° \sqrt{1685}}{1685})$

Step 7.4

Simplify the right side.

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Step 7.4.1

Evaluate $\arcsin (\frac{34 ° \sqrt{1685}}{1685})$ .

$A = 55.92280471$

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.

$A = 180 - 55.92280471$

Step 7.6

Subtract $55.92280471$ from $180$ .

$A = 124.07719528$

Step 7.7

The solution to the equation $A = 55.92280471$ .

$A = 55.92280471, 124.07719528$

Step 7.8

Exclude the invalid angle.

$A = 55.92280471$

Step 8

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

$55.92280471 + 90 ° + B = 180$

Step 9

Solve the equation for $B$ .

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Step 9.1

Add $55.92280471$ and $90 °$ .

$145.92280471 + B = 180$

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 $145.92280471$ from both sides of the equation.

$B = 180 - 145.92280471$

Step 9.2.2

Subtract $145.92280471$ from $180$ .

$B = 34.07719528$

Step 10

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

$A = 55.92280471$

$B = 34.07719528$

$C = 90 °$

$a = 34 °$

$b = 23$

$c = \sqrt{1685}$