Trigonometry Examples

$A = 122$ , $a = 24$ , $b = 24$

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 $B$ .

$\frac{\sin (B)}{24} = \frac{\sin (122)}{24}$

Step 3

Solve the equation for $B$ .

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Step 3.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$\sin (B) = \sin (122)$

Step 3.2

For the two functions to be equal, the arguments of each must be equal.

$B = 122$

Step 3.3

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.

$B = 180 - 122$

Step 3.4

Subtract $122$ from $180$ .

$B = 58$

Step 3.5

Find the period of $\sin (B)$ .

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Step 3.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 3.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 3.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 3.5.4

Divide $360$ by $1$ .

$360$

Step 3.6

The period of the $\sin (B)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$B = 58 + 360 n, 122 + 360 n$ , for any integer $n$

Step 3.7

The triangle is invalid.

Invalid triangle

Step 4

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

Step 5

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

$\frac{\sin (B)}{24} = \frac{\sin (122)}{24}$

Step 6

Solve the equation for $B$ .

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Step 6.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$\sin (B) = \sin (122)$

Step 6.2

For the two functions to be equal, the arguments of each must be equal.

$B = 122$

Step 6.3

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.

$B = 180 - 122$

Step 6.4

Subtract $122$ from $180$ .

$B = 58$

Step 6.5

Find the period of $\sin (B)$ .

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Step 6.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 6.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 6.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 6.5.4

Divide $360$ by $1$ .

$360$

Step 6.6

The period of the $\sin (B)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$B = 58 + 360 n, 122 + 360 n$ , for any integer $n$

Step 6.7

The triangle is invalid.

Invalid triangle

Step 7

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

Step 8

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

$\frac{\sin (B)}{24} = \frac{\sin (122)}{24}$

Step 9

Solve the equation for $B$ .

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Step 9.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$\sin (B) = \sin (122)$

Step 9.2

For the two functions to be equal, the arguments of each must be equal.

$B = 122$

Step 9.3

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.

$B = 180 - 122$

Step 9.4

Subtract $122$ from $180$ .

$B = 58$

Step 9.5

Find the period of $\sin (B)$ .

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Step 9.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 9.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 9.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 9.5.4

Divide $360$ by $1$ .

$360$

Step 9.6

The period of the $\sin (B)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$B = 58 + 360 n, 122 + 360 n$ , for any integer $n$

Step 9.7

The triangle is invalid.

Invalid triangle

Step 10

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

Step 11

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

$\frac{\sin (B)}{24} = \frac{\sin (122)}{24}$

Step 12

Solve the equation for $B$ .

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Step 12.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$\sin (B) = \sin (122)$

Step 12.2

For the two functions to be equal, the arguments of each must be equal.

$B = 122$

Step 12.3

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.

$B = 180 - 122$

Step 12.4

Subtract $122$ from $180$ .

$B = 58$

Step 12.5

Find the period of $\sin (B)$ .

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Step 12.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 12.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 12.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 12.5.4

Divide $360$ by $1$ .

$360$

Step 12.6

The period of the $\sin (B)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$B = 58 + 360 n, 122 + 360 n$ , for any integer $n$

Step 12.7

The triangle is invalid.

Invalid triangle

Step 13

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

Step 14

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

$\frac{\sin (B)}{24} = \frac{\sin (122)}{24}$

Step 15

Solve the equation for $B$ .

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Step 15.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$\sin (B) = \sin (122)$

Step 15.2

For the two functions to be equal, the arguments of each must be equal.

$B = 122$

Step 15.3

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.

$B = 180 - 122$

Step 15.4

Subtract $122$ from $180$ .

$B = 58$

Step 15.5

Find the period of $\sin (B)$ .

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Step 15.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 15.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 15.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 15.5.4

Divide $360$ by $1$ .

$360$

Step 15.6

The period of the $\sin (B)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$B = 58 + 360 n, 122 + 360 n$ , for any integer $n$

Step 15.7

The triangle is invalid.

Invalid triangle

Step 16

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

Step 17

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

$\frac{\sin (B)}{24} = \frac{\sin (122)}{24}$

Step 18

Solve the equation for $B$ .

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Step 18.1

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$\sin (B) = \sin (122)$

Step 18.2

For the two functions to be equal, the arguments of each must be equal.

$B = 122$

Step 18.3

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.

$B = 180 - 122$

Step 18.4

Subtract $122$ from $180$ .

$B = 58$

Step 18.5

Find the period of $\sin (B)$ .

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Step 18.5.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 18.5.2

Replace $b$ with $1$ in the formula for period.

$\frac{360}{| 1 |}$

Step 18.5.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{360}{1}$

Step 18.5.4

Divide $360$ by $1$ .

$360$

Step 18.6

The period of the $\sin (B)$ function is $360$ so values will repeat every $360$ degrees in both directions.

$B = 58 + 360 n, 122 + 360 n$ , for any integer $n$

Step 18.7

The triangle is invalid.

Invalid triangle

Step 19

There are not enough parameters given to solve the triangle.

Unknown triangle