Trigonometry Examples

$b = 8$ , $c = 10$ , $B = 30$

Step 1

The Law of Sines produces an ambiguous angle result. This means that there are $2$ angles that will correctly solve the equation. For the first triangle, use the first possible angle value.

Solve for the first triangle.

Step 2

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 3

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

$\frac{\sin (C)}{10} = \frac{\sin (30)}{8}$

Step 4

Solve the equation for $C$ .

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Step 4.1

Multiply both sides of the equation by $10$ .

$10 \frac{\sin (C)}{10} = 10 \frac{\sin (30)}{8}$

Step 4.2

Simplify both sides of the equation.

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Step 4.2.1

Simplify the left side.

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Step 4.2.1.1

Cancel the common factor of $10$ .

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Step 4.2.1.1.1

Cancel the common factor.

$10 \frac{\sin (C)}{10} = 10 \frac{\sin (30)}{8}$

Step 4.2.1.1.2

Rewrite the expression.

$\sin (C) = 10 \frac{\sin (30)}{8}$

Step 4.2.2

Simplify the right side.

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Step 4.2.2.1

Simplify $10 \frac{\sin (30)}{8}$ .

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Step 4.2.2.1.1

Cancel the common factor of $2$ .

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Step 4.2.2.1.1.1

Factor $2$ out of $10$ .

$\sin (C) = 2 (5) \frac{\sin (30)}{8}$

Step 4.2.2.1.1.2

Factor $2$ out of $8$ .

$\sin (C) = 2 \cdot 5 \frac{\sin (30)}{2 \cdot 4}$

Step 4.2.2.1.1.3

Cancel the common factor.

$\sin (C) = 2 \cdot 5 \frac{\sin (30)}{2 \cdot 4}$

Step 4.2.2.1.1.4

Rewrite the expression.

$\sin (C) = 5 \frac{\sin (30)}{4}$

Step 4.2.2.1.2

Combine $5$ and $\frac{\sin (30)}{4}$ .

$\sin (C) = \frac{5 \sin (30)}{4}$

Step 4.2.2.1.3

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

$\sin (C) = \frac{5 (\frac{1}{2})}{4}$

Step 4.2.2.1.4

Combine $5$ and $\frac{1}{2}$ .

$\sin (C) = \frac{\frac{5}{2}}{4}$

Step 4.2.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$\sin (C) = \frac{5}{2} \cdot \frac{1}{4}$

Step 4.2.2.1.6

Multiply $\frac{5}{2} \cdot \frac{1}{4}$ .

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Step 4.2.2.1.6.1

Multiply $\frac{5}{2}$ by $\frac{1}{4}$ .

$\sin (C) = \frac{5}{2 \cdot 4}$

Step 4.2.2.1.6.2

Multiply $2$ by $4$ .

$\sin (C) = \frac{5}{8}$

Step 4.3

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

$C = \arcsin (\frac{5}{8})$

Step 4.4

Simplify the right side.

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Step 4.4.1

Evaluate $\arcsin (\frac{5}{8})$ .

$C = 38.68218745$

Step 4.5

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.

$C = 180 - 38.68218745$

Step 4.6

Subtract $38.68218745$ from $180$ .

$C = 141.31781254$

Step 4.7

The solution to the equation $C = 38.68218745$ .

$C = 38.68218745, 141.31781254$

Step 5

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

$A + 38.68218745 + 30 = 180$

Step 6

Solve the equation for $A$ .

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Step 6.1

Add $38.68218745$ and $30$ .

$A + 68.68218745 = 180$

Step 6.2

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

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Step 6.2.1

Subtract $68.68218745$ from both sides of the equation.

$A = 180 - 68.68218745$

Step 6.2.2

Subtract $68.68218745$ from $180$ .

$A = 111.31781254$

Step 7

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

$a^{2} = b^{2} + c^{2} - 2 b c \cos (A)$

Step 8

Solve the equation.

$a = \sqrt{b^{2} + c^{2} - 2 b c \cos (A)}$

Step 9

Substitute the known values into the equation.

$a = \sqrt{{(8)}^{2} + {(10)}^{2} - 2 \cdot 8 \cdot 10 \cos (111.31781254)}$

Step 10

Simplify the results.

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Step 10.1

Raise $8$ to the power of $2$ .

$a = \sqrt{64 + {(10)}^{2} - 2 \cdot 8 \cdot (10 \cos (111.31781254))}$

Step 10.2

Raise $10$ to the power of $2$ .

$a = \sqrt{64 + 100 - 2 \cdot 8 \cdot (10 \cos (111.31781254))}$

Step 10.3

Multiply $- 2 \cdot 8 \cdot 10$ .

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Step 10.3.1

Multiply $- 2$ by $8$ .

$a = \sqrt{64 + 100 - 16 \cdot (10 \cos (111.31781254))}$

Step 10.3.2

Multiply $- 16$ by $10$ .

$a = \sqrt{64 + 100 - 160 \cos (111.31781254)}$

Step 10.4

Add $64$ and $100$ .

$a = \sqrt{164 - 160 \cos (111.31781254)}$

Step 10.5

Rewrite $164 - 160 \cos (111.31781254)$ as $2^{2} (41 - 40 \cos (111.31781254))$ .

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Step 10.5.1

Factor $4$ out of $164$ .

$a = \sqrt{4 (41) - 160 \cos (111.31781254)}$

Step 10.5.2

Factor $4$ out of $- 160 \cos (111.31781254)$ .

$a = \sqrt{4 (41) + 4 (- 40 \cos (111.31781254))}$

Step 10.5.3

Factor $4$ out of $4 (41) + 4 (- 40 \cos (111.31781254))$ .

$a = \sqrt{4 (41 - 40 \cos (111.31781254))}$

Step 10.5.4

Rewrite $4$ as $2^{2}$ .

$a = \sqrt{2^{2} (41 - 40 \cos (111.31781254))}$

Step 10.6

Pull terms out from under the radical.

$a = 2 \sqrt{41 - 40 \cos (111.31781254)}$

Step 11

For the second triangle, use the second possible angle value.

Solve for the second triangle.

Step 12

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

Step 13

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

$\frac{\sin (C)}{10} = \frac{\sin (30)}{8}$

Step 14

Solve the equation for $C$ .

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Step 14.1

Multiply both sides of the equation by $10$ .

$10 \frac{\sin (C)}{10} = 10 \frac{\sin (30)}{8}$

Step 14.2

Simplify both sides of the equation.

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Step 14.2.1

Simplify the left side.

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Step 14.2.1.1

Cancel the common factor of $10$ .

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Step 14.2.1.1.1

Cancel the common factor.

$10 \frac{\sin (C)}{10} = 10 \frac{\sin (30)}{8}$

Step 14.2.1.1.2

Rewrite the expression.

$\sin (C) = 10 \frac{\sin (30)}{8}$

Step 14.2.2

Simplify the right side.

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Step 14.2.2.1

Simplify $10 \frac{\sin (30)}{8}$ .

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Step 14.2.2.1.1

Cancel the common factor of $2$ .

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Step 14.2.2.1.1.1

Factor $2$ out of $10$ .

$\sin (C) = 2 (5) \frac{\sin (30)}{8}$

Step 14.2.2.1.1.2

Factor $2$ out of $8$ .

$\sin (C) = 2 \cdot 5 \frac{\sin (30)}{2 \cdot 4}$

Step 14.2.2.1.1.3

Cancel the common factor.

$\sin (C) = 2 \cdot 5 \frac{\sin (30)}{2 \cdot 4}$

Step 14.2.2.1.1.4

Rewrite the expression.

$\sin (C) = 5 \frac{\sin (30)}{4}$

Step 14.2.2.1.2

Combine $5$ and $\frac{\sin (30)}{4}$ .

$\sin (C) = \frac{5 \sin (30)}{4}$

Step 14.2.2.1.3

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

$\sin (C) = \frac{5 (\frac{1}{2})}{4}$

Step 14.2.2.1.4

Combine $5$ and $\frac{1}{2}$ .

$\sin (C) = \frac{\frac{5}{2}}{4}$

Step 14.2.2.1.5

Multiply the numerator by the reciprocal of the denominator.

$\sin (C) = \frac{5}{2} \cdot \frac{1}{4}$

Step 14.2.2.1.6

Multiply $\frac{5}{2} \cdot \frac{1}{4}$ .

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Step 14.2.2.1.6.1

Multiply $\frac{5}{2}$ by $\frac{1}{4}$ .

$\sin (C) = \frac{5}{2 \cdot 4}$

Step 14.2.2.1.6.2

Multiply $2$ by $4$ .

$\sin (C) = \frac{5}{8}$

Step 14.3

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

$C = \arcsin (\frac{5}{8})$

Step 14.4

Simplify the right side.

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Step 14.4.1

Evaluate $\arcsin (\frac{5}{8})$ .

$C = 38.68218745$

Step 14.5

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.

$C = 180 - 38.68218745$

Step 14.6

Subtract $38.68218745$ from $180$ .

$C = 141.31781254$

Step 14.7

The solution to the equation $C = 38.68218745$ .

$C = 38.68218745, 141.31781254$

Step 15

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

$A + 141.31781254 + 30 = 180$

Step 16

Solve the equation for $A$ .

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Step 16.1

Add $141.31781254$ and $30$ .

$A + 171.31781254 = 180$

Step 16.2

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

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Step 16.2.1

Subtract $171.31781254$ from both sides of the equation.

$A = 180 - 171.31781254$

Step 16.2.2

Subtract $171.31781254$ from $180$ .

$A = 8.68218745$

Step 17

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

Step 18

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

$\frac{\sin (8.68218745)}{a} = \frac{\sin (30)}{8}$

Step 19

Solve the equation for $a$ .

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Step 19.1

Factor each term.

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Step 19.1.1

Evaluate $\sin (8.68218745)$ .

$\frac{0.1509535}{a} = \frac{\sin (30)}{8}$

Step 19.1.2

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

$\frac{0.1509535}{a} = \frac{\frac{1}{2}}{8}$

Step 19.1.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{0.1509535}{a} = \frac{1}{2} \cdot \frac{1}{8}$

Step 19.1.4

Multiply $\frac{1}{2} \cdot \frac{1}{8}$ .

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Step 19.1.4.1

Multiply $\frac{1}{2}$ by $\frac{1}{8}$ .

$\frac{0.1509535}{a} = \frac{1}{2 \cdot 8}$

Step 19.1.4.2

Multiply $2$ by $8$ .

$\frac{0.1509535}{a} = \frac{1}{16}$

Step 19.2

Find the LCD of the terms in the equation.

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Step 19.2.1

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

$a, 16$

Step 19.2.2

Since $a, 16$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 16$ then find LCM for the variable part $a^{1}$ .

Step 19.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 19.2.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 19.2.5

The prime factors for $16$ are $2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 19.2.5.1

$16$ has factors of $2$ and $8$ .

$2 \cdot 8$

Step 19.2.5.2

$8$ has factors of $2$ and $4$ .

$2 \cdot 2 \cdot 4$

Step 19.2.5.3

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2 \cdot 2$

Step 19.2.6

Multiply $2 \cdot 2 \cdot 2 \cdot 2$ .

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Step 19.2.6.1

Multiply $2$ by $2$ .

$4 \cdot 2 \cdot 2$

Step 19.2.6.2

Multiply $4$ by $2$ .

$8 \cdot 2$

Step 19.2.6.3

Multiply $8$ by $2$ .

$16$

Step 19.2.7

The factor for $a^{1}$ is $a$ itself.

$a^{1} = a$

$a$ occurs $1$ time.

Step 19.2.8

The LCM of $a^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$a$

Step 19.2.9

The LCM for $a, 16$ is the numeric part $16$ multiplied by the variable part.

$16 a$

Step 19.3

Multiply each term in $\frac{0.1509535}{a} = \frac{1}{16}$ by $16 a$ to eliminate the fractions.

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Step 19.3.1

Multiply each term in $\frac{0.1509535}{a} = \frac{1}{16}$ by $16 a$ .

$\frac{0.1509535}{a} (16 a) = \frac{1}{16} (16 a)$

Step 19.3.2

Simplify the left side.

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Step 19.3.2.1

Rewrite using the commutative property of multiplication.

$16 \frac{0.1509535}{a} a = \frac{1}{16} (16 a)$

Step 19.3.2.2

Multiply $16 \frac{0.1509535}{a}$ .

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Step 19.3.2.2.1

Combine $16$ and $\frac{0.1509535}{a}$ .

$\frac{16 \cdot 0.1509535}{a} a = \frac{1}{16} (16 a)$

Step 19.3.2.2.2

Multiply $16$ by $0.1509535$ .

$\frac{2.41525603}{a} a = \frac{1}{16} (16 a)$

Step 19.3.2.3

Cancel the common factor of $a$ .

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Step 19.3.2.3.1

Cancel the common factor.

$\frac{2.41525603}{a} a = \frac{1}{16} (16 a)$

Step 19.3.2.3.2

Rewrite the expression.

$2.41525603 = \frac{1}{16} (16 a)$

Step 19.3.3

Simplify the right side.

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Step 19.3.3.1

Cancel the common factor of $16$ .

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Step 19.3.3.1.1

Factor $16$ out of $16 a$ .

$2.41525603 = \frac{1}{16} (16 (a))$

Step 19.3.3.1.2

Cancel the common factor.

$2.41525603 = \frac{1}{16} (16 a)$

Step 19.3.3.1.3

Rewrite the expression.

$2.41525603 = a$

Step 19.4

Rewrite the equation as $a = 2.41525603$ .

$a = 2.41525603$

Step 20

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

First Triangle Combination:

$A = 111.31781254$

$B = 30$

$C = 38.68218745$

$a = 2 \sqrt{41 - 40 \cos (111.31781254)}$

$b = 8$

$c = 10$

Second Triangle Combination:

$A = 8.68218745$

$B = 30$

$C = 141.31781254$

$a = 2.41525603$

$b = 8$

$c = 10$