Trigonometry Examples

$B = 143$ , $a = 30$ , $b = 28$

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 $A$ .

$\frac{\sin (A)}{30} = \frac{\sin (143)}{28}$

Step 3

Solve the equation for $A$ .

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Step 3.1

Multiply both sides of the equation by $30$ .

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Cancel the common factor of $30$ .

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Step 3.2.1.1.1

Cancel the common factor.

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (A) = 30 \frac{\sin (143)}{28}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $30 \frac{\sin (143)}{28}$ .

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Step 3.2.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.1

Factor $2$ out of $30$ .

$\sin (A) = 2 (15) \frac{\sin (143)}{28}$

Step 3.2.2.1.1.2

Factor $2$ out of $28$ .

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 3.2.2.1.1.3

Cancel the common factor.

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 3.2.2.1.1.4

Rewrite the expression.

$\sin (A) = 15 \frac{\sin (143)}{14}$

Step 3.2.2.1.2

Combine $15$ and $\frac{\sin (143)}{14}$ .

$\sin (A) = \frac{15 \sin (143)}{14}$

Step 3.2.2.1.3

Evaluate $\sin (143)$ .

$\sin (A) = \frac{15 \cdot 0.60181502}{14}$

Step 3.2.2.1.4

Multiply $15$ by $0.60181502$ .

$\sin (A) = \frac{9.02722534}{14}$

Step 3.2.2.1.5

Divide $9.02722534$ by $14$ .

$\sin (A) = 0.64480181$

Step 3.3

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

$A = \arcsin (0.64480181)$

Step 3.4

Simplify the right side.

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Step 3.4.1

Evaluate $\arcsin (0.64480181)$ .

$A = 40.15081752$

Step 3.5

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.

$A = 180 - 40.15081752$

Step 3.6

Subtract $40.15081752$ from $180$ .

$A = 139.84918247$

Step 3.7

The solution to the equation $A = 40.15081752$ .

$A = 40.15081752, 139.84918247$

Step 3.8

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 $A$ .

$\frac{\sin (A)}{30} = \frac{\sin (143)}{28}$

Step 6

Solve the equation for $A$ .

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Step 6.1

Multiply both sides of the equation by $30$ .

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 6.2

Simplify both sides of the equation.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Cancel the common factor of $30$ .

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Step 6.2.1.1.1

Cancel the common factor.

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 6.2.1.1.2

Rewrite the expression.

$\sin (A) = 30 \frac{\sin (143)}{28}$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $30 \frac{\sin (143)}{28}$ .

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Step 6.2.2.1.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1.1

Factor $2$ out of $30$ .

$\sin (A) = 2 (15) \frac{\sin (143)}{28}$

Step 6.2.2.1.1.2

Factor $2$ out of $28$ .

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 6.2.2.1.1.3

Cancel the common factor.

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 6.2.2.1.1.4

Rewrite the expression.

$\sin (A) = 15 \frac{\sin (143)}{14}$

Step 6.2.2.1.2

Combine $15$ and $\frac{\sin (143)}{14}$ .

$\sin (A) = \frac{15 \sin (143)}{14}$

Step 6.2.2.1.3

Evaluate $\sin (143)$ .

$\sin (A) = \frac{15 \cdot 0.60181502}{14}$

Step 6.2.2.1.4

Multiply $15$ by $0.60181502$ .

$\sin (A) = \frac{9.02722534}{14}$

Step 6.2.2.1.5

Divide $9.02722534$ by $14$ .

$\sin (A) = 0.64480181$

Step 6.3

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

$A = \arcsin (0.64480181)$

Step 6.4

Simplify the right side.

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Step 6.4.1

Evaluate $\arcsin (0.64480181)$ .

$A = 40.15081752$

Step 6.5

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.

$A = 180 - 40.15081752$

Step 6.6

Subtract $40.15081752$ from $180$ .

$A = 139.84918247$

Step 6.7

The solution to the equation $A = 40.15081752$ .

$A = 40.15081752, 139.84918247$

Step 6.8

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 $A$ .

$\frac{\sin (A)}{30} = \frac{\sin (143)}{28}$

Step 9

Solve the equation for $A$ .

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Step 9.1

Multiply both sides of the equation by $30$ .

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 9.2

Simplify both sides of the equation.

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Step 9.2.1

Simplify the left side.

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Step 9.2.1.1

Cancel the common factor of $30$ .

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Step 9.2.1.1.1

Cancel the common factor.

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 9.2.1.1.2

Rewrite the expression.

$\sin (A) = 30 \frac{\sin (143)}{28}$

Step 9.2.2

Simplify the right side.

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Step 9.2.2.1

Simplify $30 \frac{\sin (143)}{28}$ .

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Step 9.2.2.1.1

Cancel the common factor of $2$ .

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Step 9.2.2.1.1.1

Factor $2$ out of $30$ .

$\sin (A) = 2 (15) \frac{\sin (143)}{28}$

Step 9.2.2.1.1.2

Factor $2$ out of $28$ .

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 9.2.2.1.1.3

Cancel the common factor.

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 9.2.2.1.1.4

Rewrite the expression.

$\sin (A) = 15 \frac{\sin (143)}{14}$

Step 9.2.2.1.2

Combine $15$ and $\frac{\sin (143)}{14}$ .

$\sin (A) = \frac{15 \sin (143)}{14}$

Step 9.2.2.1.3

Evaluate $\sin (143)$ .

$\sin (A) = \frac{15 \cdot 0.60181502}{14}$

Step 9.2.2.1.4

Multiply $15$ by $0.60181502$ .

$\sin (A) = \frac{9.02722534}{14}$

Step 9.2.2.1.5

Divide $9.02722534$ by $14$ .

$\sin (A) = 0.64480181$

Step 9.3

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

$A = \arcsin (0.64480181)$

Step 9.4

Simplify the right side.

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Step 9.4.1

Evaluate $\arcsin (0.64480181)$ .

$A = 40.15081752$

Step 9.5

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.

$A = 180 - 40.15081752$

Step 9.6

Subtract $40.15081752$ from $180$ .

$A = 139.84918247$

Step 9.7

The solution to the equation $A = 40.15081752$ .

$A = 40.15081752, 139.84918247$

Step 9.8

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 $A$ .

$\frac{\sin (A)}{30} = \frac{\sin (143)}{28}$

Step 12

Solve the equation for $A$ .

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Step 12.1

Multiply both sides of the equation by $30$ .

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 12.2

Simplify both sides of the equation.

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Step 12.2.1

Simplify the left side.

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Step 12.2.1.1

Cancel the common factor of $30$ .

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Step 12.2.1.1.1

Cancel the common factor.

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 12.2.1.1.2

Rewrite the expression.

$\sin (A) = 30 \frac{\sin (143)}{28}$

Step 12.2.2

Simplify the right side.

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Step 12.2.2.1

Simplify $30 \frac{\sin (143)}{28}$ .

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Step 12.2.2.1.1

Cancel the common factor of $2$ .

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Step 12.2.2.1.1.1

Factor $2$ out of $30$ .

$\sin (A) = 2 (15) \frac{\sin (143)}{28}$

Step 12.2.2.1.1.2

Factor $2$ out of $28$ .

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 12.2.2.1.1.3

Cancel the common factor.

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 12.2.2.1.1.4

Rewrite the expression.

$\sin (A) = 15 \frac{\sin (143)}{14}$

Step 12.2.2.1.2

Combine $15$ and $\frac{\sin (143)}{14}$ .

$\sin (A) = \frac{15 \sin (143)}{14}$

Step 12.2.2.1.3

Evaluate $\sin (143)$ .

$\sin (A) = \frac{15 \cdot 0.60181502}{14}$

Step 12.2.2.1.4

Multiply $15$ by $0.60181502$ .

$\sin (A) = \frac{9.02722534}{14}$

Step 12.2.2.1.5

Divide $9.02722534$ by $14$ .

$\sin (A) = 0.64480181$

Step 12.3

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

$A = \arcsin (0.64480181)$

Step 12.4

Simplify the right side.

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Step 12.4.1

Evaluate $\arcsin (0.64480181)$ .

$A = 40.15081752$

Step 12.5

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.

$A = 180 - 40.15081752$

Step 12.6

Subtract $40.15081752$ from $180$ .

$A = 139.84918247$

Step 12.7

The solution to the equation $A = 40.15081752$ .

$A = 40.15081752, 139.84918247$

Step 12.8

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 $A$ .

$\frac{\sin (A)}{30} = \frac{\sin (143)}{28}$

Step 15

Solve the equation for $A$ .

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Step 15.1

Multiply both sides of the equation by $30$ .

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 15.2

Simplify both sides of the equation.

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Step 15.2.1

Simplify the left side.

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Step 15.2.1.1

Cancel the common factor of $30$ .

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Step 15.2.1.1.1

Cancel the common factor.

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 15.2.1.1.2

Rewrite the expression.

$\sin (A) = 30 \frac{\sin (143)}{28}$

Step 15.2.2

Simplify the right side.

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Step 15.2.2.1

Simplify $30 \frac{\sin (143)}{28}$ .

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Step 15.2.2.1.1

Cancel the common factor of $2$ .

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Step 15.2.2.1.1.1

Factor $2$ out of $30$ .

$\sin (A) = 2 (15) \frac{\sin (143)}{28}$

Step 15.2.2.1.1.2

Factor $2$ out of $28$ .

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 15.2.2.1.1.3

Cancel the common factor.

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 15.2.2.1.1.4

Rewrite the expression.

$\sin (A) = 15 \frac{\sin (143)}{14}$

Step 15.2.2.1.2

Combine $15$ and $\frac{\sin (143)}{14}$ .

$\sin (A) = \frac{15 \sin (143)}{14}$

Step 15.2.2.1.3

Evaluate $\sin (143)$ .

$\sin (A) = \frac{15 \cdot 0.60181502}{14}$

Step 15.2.2.1.4

Multiply $15$ by $0.60181502$ .

$\sin (A) = \frac{9.02722534}{14}$

Step 15.2.2.1.5

Divide $9.02722534$ by $14$ .

$\sin (A) = 0.64480181$

Step 15.3

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

$A = \arcsin (0.64480181)$

Step 15.4

Simplify the right side.

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Step 15.4.1

Evaluate $\arcsin (0.64480181)$ .

$A = 40.15081752$

Step 15.5

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.

$A = 180 - 40.15081752$

Step 15.6

Subtract $40.15081752$ from $180$ .

$A = 139.84918247$

Step 15.7

The solution to the equation $A = 40.15081752$ .

$A = 40.15081752, 139.84918247$

Step 15.8

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 $A$ .

$\frac{\sin (A)}{30} = \frac{\sin (143)}{28}$

Step 18

Solve the equation for $A$ .

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Step 18.1

Multiply both sides of the equation by $30$ .

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 18.2

Simplify both sides of the equation.

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Step 18.2.1

Simplify the left side.

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Step 18.2.1.1

Cancel the common factor of $30$ .

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Step 18.2.1.1.1

Cancel the common factor.

$30 \frac{\sin (A)}{30} = 30 \frac{\sin (143)}{28}$

Step 18.2.1.1.2

Rewrite the expression.

$\sin (A) = 30 \frac{\sin (143)}{28}$

Step 18.2.2

Simplify the right side.

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Step 18.2.2.1

Simplify $30 \frac{\sin (143)}{28}$ .

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Step 18.2.2.1.1

Cancel the common factor of $2$ .

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Step 18.2.2.1.1.1

Factor $2$ out of $30$ .

$\sin (A) = 2 (15) \frac{\sin (143)}{28}$

Step 18.2.2.1.1.2

Factor $2$ out of $28$ .

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 18.2.2.1.1.3

Cancel the common factor.

$\sin (A) = 2 \cdot 15 \frac{\sin (143)}{2 \cdot 14}$

Step 18.2.2.1.1.4

Rewrite the expression.

$\sin (A) = 15 \frac{\sin (143)}{14}$

Step 18.2.2.1.2

Combine $15$ and $\frac{\sin (143)}{14}$ .

$\sin (A) = \frac{15 \sin (143)}{14}$

Step 18.2.2.1.3

Evaluate $\sin (143)$ .

$\sin (A) = \frac{15 \cdot 0.60181502}{14}$

Step 18.2.2.1.4

Multiply $15$ by $0.60181502$ .

$\sin (A) = \frac{9.02722534}{14}$

Step 18.2.2.1.5

Divide $9.02722534$ by $14$ .

$\sin (A) = 0.64480181$

Step 18.3

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

$A = \arcsin (0.64480181)$

Step 18.4

Simplify the right side.

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Step 18.4.1

Evaluate $\arcsin (0.64480181)$ .

$A = 40.15081752$

Step 18.5

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.

$A = 180 - 40.15081752$

Step 18.6

Subtract $40.15081752$ from $180$ .

$A = 139.84918247$

Step 18.7

The solution to the equation $A = 40.15081752$ .

$A = 40.15081752, 139.84918247$

Step 18.8

The triangle is invalid.

Invalid triangle

Step 19

There are not enough parameters given to solve the triangle.

Unknown triangle