Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 7 \\ c = \\ a = \end{matrix} & \begin{matrix} A = & 15 \\ B = & 113 \\ C = \end{matrix} \end{matrix}$

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 $a$ .

$\frac{\sin (15)}{a} = \frac{\sin (113)}{7}$

Step 3

Solve the equation for $a$ .

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Step 3.1

Factor each term.

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Step 3.1.1

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 3.1.1.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{\sin (45 - 30)}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.2

Separate negation.

$\frac{\sin (45 - (30))}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.3

Apply the difference of angles identity.

$\frac{\sin (45) \cos (30) - \cos (45) \sin (30)}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.5

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

$\frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.6

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.7

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

$\frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 3.1.1.8.1

Simplify each term.

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Step 3.1.1.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 3.1.1.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.8.1.1.2

Combine using the product rule for radicals.

$\frac{\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.8.1.1.3

Multiply $2$ by $3$ .

$\frac{\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.8.1.1.4

Multiply $2$ by $2$ .

$\frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.8.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 3.1.1.8.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.8.1.2.2

Multiply $2$ by $2$ .

$\frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}}{a} = \frac{\sin (113)}{7}$

Step 3.1.1.8.2

Combine the numerators over the common denominator.

$\frac{\frac{\sqrt{6} - \sqrt{2}}{4}}{a} = \frac{\sin (113)}{7}$

Step 3.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{1}{a} = \frac{\sin (113)}{7}$

Step 3.1.3

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4}$ by $\frac{1}{a}$ .

$\frac{\sqrt{6} - \sqrt{2}}{4 a} = \frac{\sin (113)}{7}$

Step 3.1.4

Evaluate $\sin (113)$ .

$\frac{\sqrt{6} - \sqrt{2}}{4 a} = \frac{0.92050485}{7}$

Step 3.1.5

Divide $0.92050485$ by $7$ .

$\frac{\sqrt{6} - \sqrt{2}}{4 a} = 0.13150069$

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.

$4 a, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$4 a$

Step 3.3

Multiply each term in $\frac{\sqrt{6} - \sqrt{2}}{4 a} = 0.13150069$ by $4 a$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{\sqrt{6} - \sqrt{2}}{4 a} = 0.13150069$ by $4 a$ .

$\frac{\sqrt{6} - \sqrt{2}}{4 a} (4 a) = 0.13150069 (4 a)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Rewrite using the commutative property of multiplication.

$4 \frac{\sqrt{6} - \sqrt{2}}{4 a} a = 0.13150069 (4 a)$

Step 3.3.2.2

Cancel the common factor of $4$ .

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Step 3.3.2.2.1

Factor $4$ out of $4 a$ .

$4 \frac{\sqrt{6} - \sqrt{2}}{4 (a)} a = 0.13150069 (4 a)$

Step 3.3.2.2.2

Cancel the common factor.

$4 \frac{\sqrt{6} - \sqrt{2}}{4 a} a = 0.13150069 (4 a)$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{\sqrt{6} - \sqrt{2}}{a} a = 0.13150069 (4 a)$

Step 3.3.2.3

Cancel the common factor of $a$ .

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Step 3.3.2.3.1

Cancel the common factor.

$\frac{\sqrt{6} - \sqrt{2}}{a} a = 0.13150069 (4 a)$

Step 3.3.2.3.2

Rewrite the expression.

$\sqrt{6} - \sqrt{2} = 0.13150069 (4 a)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Multiply $4$ by $0.13150069$ .

$\sqrt{6} - \sqrt{2} = 0.52600277 a$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $0.52600277 a = \sqrt{6} - \sqrt{2}$ .

$0.52600277 a = \sqrt{6} - \sqrt{2}$

Step 3.4.2

Divide each term in $0.52600277 a = \sqrt{6} - \sqrt{2}$ by $0.52600277$ and simplify.

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Step 3.4.2.1

Divide each term in $0.52600277 a = \sqrt{6} - \sqrt{2}$ by $0.52600277$ .

$\frac{0.52600277 a}{0.52600277} = \frac{\sqrt{6}}{0.52600277} + \frac{- \sqrt{2}}{0.52600277}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $0.52600277$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.52600277 a}{0.52600277} = \frac{\sqrt{6}}{0.52600277} + \frac{- \sqrt{2}}{0.52600277}$

Step 3.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{\sqrt{6}}{0.52600277} + \frac{- \sqrt{2}}{0.52600277}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Simplify each term.

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Step 3.4.2.3.1.1

Evaluate the root.

$a = \frac{2.44948974}{0.52600277} + \frac{- \sqrt{2}}{0.52600277}$

Step 3.4.2.3.1.2

Divide $2.44948974$ by $0.52600277$ .

$a = 4.65680005 + \frac{- \sqrt{2}}{0.52600277}$

Step 3.4.2.3.1.3

Move the negative in front of the fraction.

$a = 4.65680005 - \frac{\sqrt{2}}{0.52600277}$

Step 3.4.2.3.1.4

Evaluate the root.

$a = 4.65680005 - \frac{1.41421356}{0.52600277}$

Step 3.4.2.3.1.5

Divide $1.41421356$ by $0.52600277$ .

$a = 4.65680005 - 1 \cdot 2.68860476$

Step 3.4.2.3.1.6

Multiply $- 1$ by $2.68860476$ .

$a = 4.65680005 - 2.68860476$

Step 3.4.2.3.2

Subtract $2.68860476$ from $4.65680005$ .

$a = 1.96819528$

Step 4

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

$15 + C + 113 = 180$

Step 5

Solve the equation for $C$ .

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Step 5.1

Add $15$ and $113$ .

$C + 128 = 180$

Step 5.2

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

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Step 5.2.1

Subtract $128$ from both sides of the equation.

$C = 180 - 128$

Step 5.2.2

Subtract $128$ from $180$ .

$C = 52$

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 $c$ .

$\frac{\sin (52)}{c} = \frac{\sin (15)}{1.96819528}$

Step 8

Solve the equation for $c$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (52)$ .

$\frac{0.78801075}{c} = \frac{\sin (15)}{1.96819528}$

Step 8.1.2

The exact value of $\sin (15)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 8.1.2.1

Split $15$ into two angles where the values of the six trigonometric functions are known.

$\frac{0.78801075}{c} = \frac{\sin (45 - 30)}{1.96819528}$

Step 8.1.2.2

Separate negation.

$\frac{0.78801075}{c} = \frac{\sin (45 - (30))}{1.96819528}$

Step 8.1.2.3

Apply the difference of angles identity.

$\frac{0.78801075}{c} = \frac{\sin (45) \cos (30) - \cos (45) \sin (30)}{1.96819528}$

Step 8.1.2.4

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{0.78801075}{c} = \frac{\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30)}{1.96819528}$

Step 8.1.2.5

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

$\frac{0.78801075}{c} = \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30)}{1.96819528}$

Step 8.1.2.6

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$\frac{0.78801075}{c} = \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30)}{1.96819528}$

Step 8.1.2.7

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

$\frac{0.78801075}{c} = \frac{\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{1.96819528}$

Step 8.1.2.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 8.1.2.8.1

Simplify each term.

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Step 8.1.2.8.1.1

Multiply $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}$ .

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Step 8.1.2.8.1.1.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{\sqrt{3}}{2}$ .

$\frac{0.78801075}{c} = \frac{\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{1.96819528}$

Step 8.1.2.8.1.1.2

Combine using the product rule for radicals.

$\frac{0.78801075}{c} = \frac{\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{1.96819528}$

Step 8.1.2.8.1.1.3

Multiply $2$ by $3$ .

$\frac{0.78801075}{c} = \frac{\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{1.96819528}$

Step 8.1.2.8.1.1.4

Multiply $2$ by $2$ .

$\frac{0.78801075}{c} = \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}}{1.96819528}$

Step 8.1.2.8.1.2

Multiply $- \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 8.1.2.8.1.2.1

Multiply $\frac{1}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{0.78801075}{c} = \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}}{1.96819528}$

Step 8.1.2.8.1.2.2

Multiply $2$ by $2$ .

$\frac{0.78801075}{c} = \frac{\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}}{1.96819528}$

Step 8.1.2.8.2

Combine the numerators over the common denominator.

$\frac{0.78801075}{c} = \frac{\frac{\sqrt{6} - \sqrt{2}}{4}}{1.96819528}$

Step 8.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.78801075}{c} = \frac{\sqrt{6} - \sqrt{2}}{4} \cdot \frac{1}{1.96819528}$

Step 8.1.4

Divide $1$ by $1.96819528$ .

$\frac{0.78801075}{c} = \frac{\sqrt{6} - \sqrt{2}}{4} \cdot 0.50807966$

Step 8.1.5

Multiply $\frac{\sqrt{6} - \sqrt{2}}{4} \cdot 0.50807966$ .

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Step 8.1.5.1

Combine $\frac{\sqrt{6} - \sqrt{2}}{4}$ and $0.50807966$ .

$\frac{0.78801075}{c} = \frac{(\sqrt{6} - \sqrt{2}) \cdot 0.50807966}{4}$

Step 8.1.5.2

Multiply $\sqrt{6} - \sqrt{2}$ by $0.50807966$ .

$\frac{0.78801075}{c} = \frac{0.52600277}{4}$

Step 8.1.6

Divide $0.52600277$ by $4$ .

$\frac{0.78801075}{c} = 0.13150069$

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.

$c, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$c$

Step 8.3

Multiply each term in $\frac{0.78801075}{c} = 0.13150069$ by $c$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.78801075}{c} = 0.13150069$ by $c$ .

$\frac{0.78801075}{c} c = 0.13150069 c$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $c$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.78801075}{c} c = 0.13150069 c$

Step 8.3.2.1.2

Rewrite the expression.

$0.78801075 = 0.13150069 c$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.13150069 c = 0.78801075$ .

$0.13150069 c = 0.78801075$

Step 8.4.2

Divide each term in $0.13150069 c = 0.78801075$ by $0.13150069$ and simplify.

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Step 8.4.2.1

Divide each term in $0.13150069 c = 0.78801075$ by $0.13150069$ .

$\frac{0.13150069 c}{0.13150069} = \frac{0.78801075}{0.13150069}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.13150069$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.13150069 c}{0.13150069} = \frac{0.78801075}{0.13150069}$

Step 8.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{0.78801075}{0.13150069}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.78801075$ by $0.13150069$ .

$c = 5.99244561$

Step 9

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

$A = 15$

$B = 113$

$C = 52$

$a = 1.96819528$

$b = 7$

$c = 5.99244561$