Trigonometry Examples

$A = 40$ , $C = 60$ , $a = 6$

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 $c$ .

$\frac{\sin (60)}{c} = \frac{\sin (40)}{6}$

Step 3

Solve the equation for $c$ .

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Step 3.1

Factor each term.

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Step 3.1.1

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

$\frac{\frac{\sqrt{3}}{2}}{c} = \frac{\sin (40)}{6}$

Step 3.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{3}}{2} \cdot \frac{1}{c} = \frac{\sin (40)}{6}$

Step 3.1.3

Multiply $\frac{\sqrt{3}}{2}$ by $\frac{1}{c}$ .

$\frac{\sqrt{3}}{2 c} = \frac{\sin (40)}{6}$

Step 3.1.4

Evaluate $\sin (40)$ .

$\frac{\sqrt{3}}{2 c} = \frac{0.6427876}{6}$

Step 3.1.5

Divide $0.6427876$ by $6$ .

$\frac{\sqrt{3}}{2 c} = 0.10713126$

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.

$2 c, 1$

Step 3.2.2

The LCM of one and any expression is the expression.

$2 c$

Step 3.3

Multiply each term in $\frac{\sqrt{3}}{2 c} = 0.10713126$ by $2 c$ to eliminate the fractions.

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Step 3.3.1

Multiply each term in $\frac{\sqrt{3}}{2 c} = 0.10713126$ by $2 c$ .

$\frac{\sqrt{3}}{2 c} (2 c) = 0.10713126 (2 c)$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Rewrite using the commutative property of multiplication.

$2 \frac{\sqrt{3}}{2 c} c = 0.10713126 (2 c)$

Step 3.3.2.2

Cancel the common factor of $2$ .

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Step 3.3.2.2.1

Factor $2$ out of $2 c$ .

$2 \frac{\sqrt{3}}{2 (c)} c = 0.10713126 (2 c)$

Step 3.3.2.2.2

Cancel the common factor.

$2 \frac{\sqrt{3}}{2 c} c = 0.10713126 (2 c)$

Step 3.3.2.2.3

Rewrite the expression.

$\frac{\sqrt{3}}{c} c = 0.10713126 (2 c)$

Step 3.3.2.3

Cancel the common factor of $c$ .

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Step 3.3.2.3.1

Cancel the common factor.

$\frac{\sqrt{3}}{c} c = 0.10713126 (2 c)$

Step 3.3.2.3.2

Rewrite the expression.

$\sqrt{3} = 0.10713126 (2 c)$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Multiply $2$ by $0.10713126$ .

$\sqrt{3} = 0.21426253 c$

Step 3.4

Solve the equation.

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Step 3.4.1

Rewrite the equation as $0.21426253 c = \sqrt{3}$ .

$0.21426253 c = \sqrt{3}$

Step 3.4.2

Divide each term in $0.21426253 c = \sqrt{3}$ by $0.21426253$ and simplify.

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Step 3.4.2.1

Divide each term in $0.21426253 c = \sqrt{3}$ by $0.21426253$ .

$\frac{0.21426253 c}{0.21426253} = \frac{\sqrt{3}}{0.21426253}$

Step 3.4.2.2

Simplify the left side.

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Step 3.4.2.2.1

Cancel the common factor of $0.21426253$ .

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Step 3.4.2.2.1.1

Cancel the common factor.

$\frac{0.21426253 c}{0.21426253} = \frac{\sqrt{3}}{0.21426253}$

Step 3.4.2.2.1.2

Divide $c$ by $1$ .

$c = \frac{\sqrt{3}}{0.21426253}$

Step 3.4.2.3

Simplify the right side.

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Step 3.4.2.3.1

Evaluate the root.

$c = \frac{1.7320508}{0.21426253}$

Step 3.4.2.3.2

Divide $1.7320508$ by $0.21426253$ .

$c = 8.08377813$

Step 4

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

$40 + 60 + B = 180$

Step 5

Solve the equation for $B$ .

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Step 5.1

Add $40$ and $60$ .

$100 + B = 180$

Step 5.2

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

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Step 5.2.1

Subtract $100$ from both sides of the equation.

$B = 180 - 100$

Step 5.2.2

Subtract $100$ from $180$ .

$B = 80$

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 $b$ .

$\frac{\sin (80)}{b} = \frac{\sin (40)}{6}$

Step 8

Solve the equation for $b$ .

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Step 8.1

Factor each term.

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Step 8.1.1

Evaluate $\sin (80)$ .

$\frac{0.98480775}{b} = \frac{\sin (40)}{6}$

Step 8.1.2

Evaluate $\sin (40)$ .

$\frac{0.98480775}{b} = \frac{0.6427876}{6}$

Step 8.1.3

Divide $0.6427876$ by $6$ .

$\frac{0.98480775}{b} = 0.10713126$

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.

$b, 1$

Step 8.2.2

The LCM of one and any expression is the expression.

$b$

Step 8.3

Multiply each term in $\frac{0.98480775}{b} = 0.10713126$ by $b$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.98480775}{b} = 0.10713126$ by $b$ .

$\frac{0.98480775}{b} b = 0.10713126 b$

Step 8.3.2

Simplify the left side.

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Step 8.3.2.1

Cancel the common factor of $b$ .

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Step 8.3.2.1.1

Cancel the common factor.

$\frac{0.98480775}{b} b = 0.10713126 b$

Step 8.3.2.1.2

Rewrite the expression.

$0.98480775 = 0.10713126 b$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.10713126 b = 0.98480775$ .

$0.10713126 b = 0.98480775$

Step 8.4.2

Divide each term in $0.10713126 b = 0.98480775$ by $0.10713126$ and simplify.

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Step 8.4.2.1

Divide each term in $0.10713126 b = 0.98480775$ by $0.10713126$ .

$\frac{0.10713126 b}{0.10713126} = \frac{0.98480775}{0.10713126}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.10713126$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.10713126 b}{0.10713126} = \frac{0.98480775}{0.10713126}$

Step 8.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.98480775}{0.10713126}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.98480775$ by $0.10713126$ .

$b = 9.19253331$

Step 9

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

$A = 40$

$B = 80$

$C = 60$

$a = 6$

$b = 9.19253331$

$c = 8.08377813$

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