Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = \\ c = & 5 \\ a = \end{matrix} & \begin{matrix} A = \\ B = & 100 \\ C = & 25 \end{matrix} \end{matrix}$

Step 1

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 2

Substitute the known values into the law of sines to find $b$ .

$\frac{\sin (100)}{b} = \frac{\sin (25)}{5}$

Step 3

Solve the equation for $b$ .

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Step 3.1

Factor each term.

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Step 3.1.1

Evaluate $\sin (100)$ .

$\frac{0.98480775}{b} = \frac{\sin (25)}{5}$

Step 3.1.2

Evaluate $\sin (25)$ .

$\frac{0.98480775}{b} = \frac{0.42261826}{5}$

Step 3.1.3

Divide $0.42261826$ by $5$ .

$\frac{0.98480775}{b} = 0.08452365$

Step 3.2

Find the LCD of the terms in the equation.

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Step 3.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$b, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$b$

Step 3.3

Multiply each term in $\frac{0.98480775}{b} = 0.08452365$ by $b$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{0.98480775}{b} = 0.08452365$ by $b$ .

$\frac{0.98480775}{b} b = 0.08452365 b$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $b$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{0.98480775}{b} b = 0.08452365 b$

Step 3.3.2.1.2

Rewrite the expression.

$0.98480775 = 0.08452365 b$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $0.08452365 b = 0.98480775$ .

$0.08452365 b = 0.98480775$

Step 3.4.2

Divide each term in $0.08452365 b = 0.98480775$ by $0.08452365$ and simplify.

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Step 3.4.2.1

Divide each term in $0.08452365 b = 0.98480775$ by $0.08452365$ .

$\frac{0.08452365 b}{0.08452365} = \frac{0.98480775}{0.08452365}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $0.08452365$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.08452365 b}{0.08452365} = \frac{0.98480775}{0.08452365}$

Step 3.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.98480775}{0.08452365}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Divide $0.98480775$ by $0.08452365$ .

$b = 11.65126832$

Step 4

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

$A + 25 + 100 = 180$

Step 5

Solve the equation for $A$ .

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Step 5.1

Add $25$ and $100$ .

$A + 125 = 180$

Step 5.2

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

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Step 5.2.1

Subtract $125$ from both sides of the equation.

$A = 180 - 125$

Step 5.2.2

Subtract $125$ from $180$ .

$A = 55$

Step 6

$\frac{\sin (A)}{a} = \frac{\sin (B)}{b} = \frac{\sin (C)}{c}$

Step 7

Substitute the known values into the law of sines to find $a$ .

$\frac{\sin (55)}{a} = \frac{\sin (100)}{11.65126832}$

Step 8

Solve the equation for $a$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (55)$ .

$\frac{0.81915204}{a} = \frac{\sin (100)}{11.65126832}$

Step 8.1.2

Evaluate $\sin (100)$ .

$\frac{0.81915204}{a} = \frac{0.98480775}{11.65126832}$

Step 8.1.3

Divide $0.98480775$ by $11.65126832$ .

$\frac{0.81915204}{a} = 0.08452365$

Step 8.2

Find the LCD of the terms in the equation.

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Step 8.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$a, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$a$

Step 8.3

Multiply each term in $\frac{0.81915204}{a} = 0.08452365$ by $a$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.81915204}{a} = 0.08452365$ by $a$ .

$\frac{0.81915204}{a} a = 0.08452365 a$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $a$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.81915204}{a} a = 0.08452365 a$

Step 8.3.2.1.2

Rewrite the expression.

$0.81915204 = 0.08452365 a$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.08452365 a = 0.81915204$ .

$0.08452365 a = 0.81915204$

Step 8.4.2

Divide each term in $0.08452365 a = 0.81915204$ by $0.08452365$ and simplify.

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Step 8.4.2.1

Divide each term in $0.08452365 a = 0.81915204$ by $0.08452365$ .

$\frac{0.08452365 a}{0.08452365} = \frac{0.81915204}{0.08452365}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.08452365$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.08452365 a}{0.08452365} = \frac{0.81915204}{0.08452365}$

Step 8.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{0.81915204}{0.08452365}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.81915204$ by $0.08452365$ .

$a = 9.69139432$