Trigonometry Examples

$A = 60$ , $B = 40$ , $c = 15$

Step 1

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

$60 + C + 40 = 180$

Step 2

Solve the equation for $C$ .

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Step 2.1

Add $60$ and $40$ .

$C + 100 = 180$

Step 2.2

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

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Step 2.2.1

Subtract $100$ from both sides of the equation.

$C = 180 - 100$

Step 2.2.2

Subtract $100$ from $180$ .

$C = 80$

Step 3

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 4

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

$\frac{\sin (60)}{a} = \frac{\sin (80)}{15}$

Step 5

Solve the equation for $a$ .

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Step 5.1

Factor each term.

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Step 5.1.1

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

$\frac{\frac{\sqrt{3}}{2}}{a} = \frac{\sin (80)}{15}$

Step 5.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sqrt{3}}{2} \cdot \frac{1}{a} = \frac{\sin (80)}{15}$

Step 5.1.3

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

$\frac{\sqrt{3}}{2 a} = \frac{\sin (80)}{15}$

Step 5.1.4

Evaluate $\sin (80)$ .

$\frac{\sqrt{3}}{2 a} = \frac{0.98480775}{15}$

Step 5.1.5

Divide $0.98480775$ by $15$ .

$\frac{\sqrt{3}}{2 a} = 0.06565385$

Step 5.2

Find the LCD of the terms in the equation.

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Step 5.2.1

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

$2 a, 1$

Step 5.2.2

The LCM of one and any expression is the expression.

$2 a$

Step 5.3

Multiply each term in $\frac{\sqrt{3}}{2 a} = 0.06565385$ by $2 a$ to eliminate the fractions.

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Step 5.3.1

Multiply each term in $\frac{\sqrt{3}}{2 a} = 0.06565385$ by $2 a$ .

$\frac{\sqrt{3}}{2 a} (2 a) = 0.06565385 (2 a)$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Rewrite using the commutative property of multiplication.

$2 \frac{\sqrt{3}}{2 a} a = 0.06565385 (2 a)$

Step 5.3.2.2

Cancel the common factor of $2$ .

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Step 5.3.2.2.1

Factor $2$ out of $2 a$ .

$2 \frac{\sqrt{3}}{2 (a)} a = 0.06565385 (2 a)$

Step 5.3.2.2.2

Cancel the common factor.

$2 \frac{\sqrt{3}}{2 a} a = 0.06565385 (2 a)$

Step 5.3.2.2.3

Rewrite the expression.

$\frac{\sqrt{3}}{a} a = 0.06565385 (2 a)$

Step 5.3.2.3

Cancel the common factor of $a$ .

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Step 5.3.2.3.1

Cancel the common factor.

$\frac{\sqrt{3}}{a} a = 0.06565385 (2 a)$

Step 5.3.2.3.2

Rewrite the expression.

$\sqrt{3} = 0.06565385 (2 a)$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Multiply $2$ by $0.06565385$ .

$\sqrt{3} = 0.1313077 a$

Step 5.4

Solve the equation.

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Step 5.4.1

Rewrite the equation as $0.1313077 a = \sqrt{3}$ .

$0.1313077 a = \sqrt{3}$

Step 5.4.2

Divide each term in $0.1313077 a = \sqrt{3}$ by $0.1313077$ and simplify.

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Step 5.4.2.1

Divide each term in $0.1313077 a = \sqrt{3}$ by $0.1313077$ .

$\frac{0.1313077 a}{0.1313077} = \frac{\sqrt{3}}{0.1313077}$

Step 5.4.2.2

Simplify the left side.

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Step 5.4.2.2.1

Cancel the common factor of $0.1313077$ .

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Step 5.4.2.2.1.1

Cancel the common factor.

$\frac{0.1313077 a}{0.1313077} = \frac{\sqrt{3}}{0.1313077}$

Step 5.4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{\sqrt{3}}{0.1313077}$

Step 5.4.2.3

Simplify the right side.

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Step 5.4.2.3.1

Evaluate the root.

$a = \frac{1.7320508}{0.1313077}$

Step 5.4.2.3.2

Divide $1.7320508$ by $0.1313077$ .

$a = 13.19077862$

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 $b$ .

$\frac{\sin (40)}{b} = \frac{\sin (60)}{13.19077862}$

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 $\sin (40)$ .

$\frac{0.6427876}{b} = \frac{\sin (60)}{13.19077862}$

Step 8.1.2

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

$\frac{0.6427876}{b} = \frac{\frac{\sqrt{3}}{2}}{13.19077862}$

Step 8.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.6427876}{b} = \frac{\sqrt{3}}{2} \cdot \frac{1}{13.19077862}$

Step 8.1.4

Divide $1$ by $13.19077862$ .

$\frac{0.6427876}{b} = \frac{\sqrt{3}}{2} \cdot 0.07581053$

Step 8.1.5

Multiply $\frac{\sqrt{3}}{2} \cdot 0.07581053$ .

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Step 8.1.5.1

Combine $\frac{\sqrt{3}}{2}$ and $0.07581053$ .

$\frac{0.6427876}{b} = \frac{\sqrt{3} \cdot 0.07581053}{2}$

Step 8.1.5.2

Multiply $\sqrt{3}$ by $0.07581053$ .

$\frac{0.6427876}{b} = \frac{0.1313077}{2}$

Step 8.1.6

Divide $0.1313077$ by $2$ .

$\frac{0.6427876}{b} = 0.06565385$

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 $\frac{0.6427876}{b} = 0.06565385$ by $b$ to eliminate the fractions.

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Step 8.3.1

Multiply each term in $\frac{0.6427876}{b} = 0.06565385$ by $b$ .

$\frac{0.6427876}{b} b = 0.06565385 b$

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.

$\frac{0.6427876}{b} b = 0.06565385 b$

Step 8.3.2.1.2

Rewrite the expression.

$0.6427876 = 0.06565385 b$

Step 8.4

Solve the equation.

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Step 8.4.1

Rewrite the equation as $0.06565385 b = 0.6427876$ .

$0.06565385 b = 0.6427876$

Step 8.4.2

Divide each term in $0.06565385 b = 0.6427876$ by $0.06565385$ and simplify.

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Step 8.4.2.1

Divide each term in $0.06565385 b = 0.6427876$ by $0.06565385$ .

$\frac{0.06565385 b}{0.06565385} = \frac{0.6427876}{0.06565385}$

Step 8.4.2.2

Simplify the left side.

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Step 8.4.2.2.1

Cancel the common factor of $0.06565385$ .

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Step 8.4.2.2.1.1

Cancel the common factor.

$\frac{0.06565385 b}{0.06565385} = \frac{0.6427876}{0.06565385}$

Step 8.4.2.2.1.2

Divide $b$ by $1$ .

$b = \frac{0.6427876}{0.06565385}$

Step 8.4.2.3

Simplify the right side.

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Step 8.4.2.3.1

Divide $0.6427876$ by $0.06565385$ .

$b = 9.79055466$

Step 9

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

$A = 60$

$B = 40$

$C = 80$

$a = 13.19077862$

$b = 9.79055466$

$c = 15$