Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 54 \\ c = & 42 \\ a = \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 52.1 \end{matrix} \end{matrix}$

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 $B$ .

$\frac{\sin (B)}{54} = \frac{\sin (52.1)}{42}$

Step 3

Solve the equation for $B$ .

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Step 3.1

Multiply both sides of the equation by $54$ .

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

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 $54$ .

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Step 3.2.1.1.1

Cancel the common factor.

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (B) = 54 \frac{\sin (52.1)}{42}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $54 \frac{\sin (52.1)}{42}$ .

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Step 3.2.2.1.1

Cancel the common factor of $6$ .

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Step 3.2.2.1.1.1

Factor $6$ out of $54$ .

$\sin (B) = 6 (9) \frac{\sin (52.1)}{42}$

Step 3.2.2.1.1.2

Factor $6$ out of $42$ .

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 3.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 3.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 9 \frac{\sin (52.1)}{7}$

Step 3.2.2.1.2

Combine $9$ and $\frac{\sin (52.1)}{7}$ .

$\sin (B) = \frac{9 \sin (52.1)}{7}$

Step 3.2.2.1.3

Evaluate $\sin (52.1)$ .

$\sin (B) = \frac{9 \cdot 0.78908408}{7}$

Step 3.2.2.1.4

Multiply $9$ by $0.78908408$ .

$\sin (B) = \frac{7.10175676}{7}$

Step 3.2.2.1.5

Divide $7.10175676$ by $7$ .

$\sin (B) = 1.01453668$

Step 3.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.01453668$ does not fall in this range, there is no solution.

No solution

Step 4

There are not enough parameters given to solve the triangle.

Unknown triangle

Step 5

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

Step 6

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

$\frac{\sin (B)}{54} = \frac{\sin (52.1)}{42}$

Step 7

Solve the equation for $B$ .

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Step 7.1

Multiply both sides of the equation by $54$ .

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

Step 7.2

Simplify both sides of the equation.

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Step 7.2.1

Simplify the left side.

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Step 7.2.1.1

Cancel the common factor of $54$ .

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Step 7.2.1.1.1

Cancel the common factor.

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (B) = 54 \frac{\sin (52.1)}{42}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $54 \frac{\sin (52.1)}{42}$ .

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Step 7.2.2.1.1

Cancel the common factor of $6$ .

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Step 7.2.2.1.1.1

Factor $6$ out of $54$ .

$\sin (B) = 6 (9) \frac{\sin (52.1)}{42}$

Step 7.2.2.1.1.2

Factor $6$ out of $42$ .

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 7.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 7.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 9 \frac{\sin (52.1)}{7}$

Step 7.2.2.1.2

Combine $9$ and $\frac{\sin (52.1)}{7}$ .

$\sin (B) = \frac{9 \sin (52.1)}{7}$

Step 7.2.2.1.3

Evaluate $\sin (52.1)$ .

$\sin (B) = \frac{9 \cdot 0.78908408}{7}$

Step 7.2.2.1.4

Multiply $9$ by $0.78908408$ .

$\sin (B) = \frac{7.10175676}{7}$

Step 7.2.2.1.5

Divide $7.10175676$ by $7$ .

$\sin (B) = 1.01453668$

Step 7.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.01453668$ does not fall in this range, there is no solution.

No solution

Step 8

There are not enough parameters given to solve the triangle.

Unknown triangle

Step 9

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

Step 10

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

$\frac{\sin (B)}{54} = \frac{\sin (52.1)}{42}$

Step 11

Solve the equation for $B$ .

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Step 11.1

Multiply both sides of the equation by $54$ .

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

Step 11.2

Simplify both sides of the equation.

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Step 11.2.1

Simplify the left side.

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Step 11.2.1.1

Cancel the common factor of $54$ .

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Step 11.2.1.1.1

Cancel the common factor.

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

Step 11.2.1.1.2

Rewrite the expression.

$\sin (B) = 54 \frac{\sin (52.1)}{42}$

Step 11.2.2

Simplify the right side.

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Step 11.2.2.1

Simplify $54 \frac{\sin (52.1)}{42}$ .

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Step 11.2.2.1.1

Cancel the common factor of $6$ .

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Step 11.2.2.1.1.1

Factor $6$ out of $54$ .

$\sin (B) = 6 (9) \frac{\sin (52.1)}{42}$

Step 11.2.2.1.1.2

Factor $6$ out of $42$ .

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 11.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 11.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 9 \frac{\sin (52.1)}{7}$

Step 11.2.2.1.2

Combine $9$ and $\frac{\sin (52.1)}{7}$ .

$\sin (B) = \frac{9 \sin (52.1)}{7}$

Step 11.2.2.1.3

Evaluate $\sin (52.1)$ .

$\sin (B) = \frac{9 \cdot 0.78908408}{7}$

Step 11.2.2.1.4

Multiply $9$ by $0.78908408$ .

$\sin (B) = \frac{7.10175676}{7}$

Step 11.2.2.1.5

Divide $7.10175676$ by $7$ .

$\sin (B) = 1.01453668$

Step 11.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.01453668$ does not fall in this range, there is no solution.

No solution

Step 12

There are not enough parameters given to solve the triangle.

Unknown 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 $B$ .

$\frac{\sin (B)}{54} = \frac{\sin (52.1)}{42}$

Step 15

Solve the equation for $B$ .

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Step 15.1

Multiply both sides of the equation by $54$ .

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

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 $54$ .

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Step 15.2.1.1.1

Cancel the common factor.

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

Step 15.2.1.1.2

Rewrite the expression.

$\sin (B) = 54 \frac{\sin (52.1)}{42}$

Step 15.2.2

Simplify the right side.

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Step 15.2.2.1

Simplify $54 \frac{\sin (52.1)}{42}$ .

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Step 15.2.2.1.1

Cancel the common factor of $6$ .

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Step 15.2.2.1.1.1

Factor $6$ out of $54$ .

$\sin (B) = 6 (9) \frac{\sin (52.1)}{42}$

Step 15.2.2.1.1.2

Factor $6$ out of $42$ .

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 15.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 15.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 9 \frac{\sin (52.1)}{7}$

Step 15.2.2.1.2

Combine $9$ and $\frac{\sin (52.1)}{7}$ .

$\sin (B) = \frac{9 \sin (52.1)}{7}$

Step 15.2.2.1.3

Evaluate $\sin (52.1)$ .

$\sin (B) = \frac{9 \cdot 0.78908408}{7}$

Step 15.2.2.1.4

Multiply $9$ by $0.78908408$ .

$\sin (B) = \frac{7.10175676}{7}$

Step 15.2.2.1.5

Divide $7.10175676$ by $7$ .

$\sin (B) = 1.01453668$

Step 15.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.01453668$ does not fall in this range, there is no solution.

No solution

Step 16

There are not enough parameters given to solve the triangle.

Unknown triangle

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 $B$ .

$\frac{\sin (B)}{54} = \frac{\sin (52.1)}{42}$

Step 19

Solve the equation for $B$ .

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Step 19.1

Multiply both sides of the equation by $54$ .

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

Step 19.2

Simplify both sides of the equation.

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Step 19.2.1

Simplify the left side.

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Step 19.2.1.1

Cancel the common factor of $54$ .

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Step 19.2.1.1.1

Cancel the common factor.

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

Step 19.2.1.1.2

Rewrite the expression.

$\sin (B) = 54 \frac{\sin (52.1)}{42}$

Step 19.2.2

Simplify the right side.

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Step 19.2.2.1

Simplify $54 \frac{\sin (52.1)}{42}$ .

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Step 19.2.2.1.1

Cancel the common factor of $6$ .

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Step 19.2.2.1.1.1

Factor $6$ out of $54$ .

$\sin (B) = 6 (9) \frac{\sin (52.1)}{42}$

Step 19.2.2.1.1.2

Factor $6$ out of $42$ .

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 19.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 19.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 9 \frac{\sin (52.1)}{7}$

Step 19.2.2.1.2

Combine $9$ and $\frac{\sin (52.1)}{7}$ .

$\sin (B) = \frac{9 \sin (52.1)}{7}$

Step 19.2.2.1.3

Evaluate $\sin (52.1)$ .

$\sin (B) = \frac{9 \cdot 0.78908408}{7}$

Step 19.2.2.1.4

Multiply $9$ by $0.78908408$ .

$\sin (B) = \frac{7.10175676}{7}$

Step 19.2.2.1.5

Divide $7.10175676$ by $7$ .

$\sin (B) = 1.01453668$

Step 19.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.01453668$ does not fall in this range, there is no solution.

No solution

Step 20

There are not enough parameters given to solve the triangle.

Unknown triangle

Step 21

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

Step 22

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

$\frac{\sin (B)}{54} = \frac{\sin (52.1)}{42}$

Step 23

Solve the equation for $B$ .

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Step 23.1

Multiply both sides of the equation by $54$ .

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

Step 23.2

Simplify both sides of the equation.

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Step 23.2.1

Simplify the left side.

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Step 23.2.1.1

Cancel the common factor of $54$ .

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Step 23.2.1.1.1

Cancel the common factor.

$54 \frac{\sin (B)}{54} = 54 \frac{\sin (52.1)}{42}$

Step 23.2.1.1.2

Rewrite the expression.

$\sin (B) = 54 \frac{\sin (52.1)}{42}$

Step 23.2.2

Simplify the right side.

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Step 23.2.2.1

Simplify $54 \frac{\sin (52.1)}{42}$ .

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Step 23.2.2.1.1

Cancel the common factor of $6$ .

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Step 23.2.2.1.1.1

Factor $6$ out of $54$ .

$\sin (B) = 6 (9) \frac{\sin (52.1)}{42}$

Step 23.2.2.1.1.2

Factor $6$ out of $42$ .

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 23.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 6 \cdot 9 \frac{\sin (52.1)}{6 \cdot 7}$

Step 23.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 9 \frac{\sin (52.1)}{7}$

Step 23.2.2.1.2

Combine $9$ and $\frac{\sin (52.1)}{7}$ .

$\sin (B) = \frac{9 \sin (52.1)}{7}$

Step 23.2.2.1.3

Evaluate $\sin (52.1)$ .

$\sin (B) = \frac{9 \cdot 0.78908408}{7}$

Step 23.2.2.1.4

Multiply $9$ by $0.78908408$ .

$\sin (B) = \frac{7.10175676}{7}$

Step 23.2.2.1.5

Divide $7.10175676$ by $7$ .

$\sin (B) = 1.01453668$

Step 23.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.01453668$ does not fall in this range, there is no solution.

No solution

Step 24

There are not enough parameters given to solve the triangle.

Unknown triangle