Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 234 \\ c = \\ a = & 178 \end{matrix} & \begin{matrix} A = & 52 \\ B = \\ C = \end{matrix} \end{matrix}$

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 $B$ .

$\frac{\sin (B)}{234} = \frac{\sin (52)}{178}$

Step 3

Solve the equation for $B$ .

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Step 3.1

Multiply both sides of the equation by $234$ .

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

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 $234$ .

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Step 3.2.1.1.1

Cancel the common factor.

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (B) = 234 \frac{\sin (52)}{178}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $234 \frac{\sin (52)}{178}$ .

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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 $234$ .

$\sin (B) = 2 (117) \frac{\sin (52)}{178}$

Step 3.2.2.1.1.2

Factor $2$ out of $178$ .

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 3.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 3.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 117 \frac{\sin (52)}{89}$

Step 3.2.2.1.2

Combine $117$ and $\frac{\sin (52)}{89}$ .

$\sin (B) = \frac{117 \sin (52)}{89}$

Step 3.2.2.1.3

Evaluate $\sin (52)$ .

$\sin (B) = \frac{117 \cdot 0.78801075}{89}$

Step 3.2.2.1.4

Multiply $117$ by $0.78801075$ .

$\sin (B) = \frac{92.19725817}{89}$

Step 3.2.2.1.5

Divide $92.19725817$ by $89$ .

$\sin (B) = 1.03592424$

Step 3.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.03592424$ 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 $B$ .

$\frac{\sin (B)}{234} = \frac{\sin (52)}{178}$

Step 7

Solve the equation for $B$ .

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Step 7.1

Multiply both sides of the equation by $234$ .

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

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 $234$ .

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Step 7.2.1.1.1

Cancel the common factor.

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (B) = 234 \frac{\sin (52)}{178}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $234 \frac{\sin (52)}{178}$ .

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Step 7.2.2.1.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.1.1

Factor $2$ out of $234$ .

$\sin (B) = 2 (117) \frac{\sin (52)}{178}$

Step 7.2.2.1.1.2

Factor $2$ out of $178$ .

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 7.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 7.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 117 \frac{\sin (52)}{89}$

Step 7.2.2.1.2

Combine $117$ and $\frac{\sin (52)}{89}$ .

$\sin (B) = \frac{117 \sin (52)}{89}$

Step 7.2.2.1.3

Evaluate $\sin (52)$ .

$\sin (B) = \frac{117 \cdot 0.78801075}{89}$

Step 7.2.2.1.4

Multiply $117$ by $0.78801075$ .

$\sin (B) = \frac{92.19725817}{89}$

Step 7.2.2.1.5

Divide $92.19725817$ by $89$ .

$\sin (B) = 1.03592424$

Step 7.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.03592424$ 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 $B$ .

$\frac{\sin (B)}{234} = \frac{\sin (52)}{178}$

Step 11

Solve the equation for $B$ .

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Step 11.1

Multiply both sides of the equation by $234$ .

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

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 $234$ .

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Step 11.2.1.1.1

Cancel the common factor.

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

Step 11.2.1.1.2

Rewrite the expression.

$\sin (B) = 234 \frac{\sin (52)}{178}$

Step 11.2.2

Simplify the right side.

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Step 11.2.2.1

Simplify $234 \frac{\sin (52)}{178}$ .

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Step 11.2.2.1.1

Cancel the common factor of $2$ .

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Step 11.2.2.1.1.1

Factor $2$ out of $234$ .

$\sin (B) = 2 (117) \frac{\sin (52)}{178}$

Step 11.2.2.1.1.2

Factor $2$ out of $178$ .

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 11.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 11.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 117 \frac{\sin (52)}{89}$

Step 11.2.2.1.2

Combine $117$ and $\frac{\sin (52)}{89}$ .

$\sin (B) = \frac{117 \sin (52)}{89}$

Step 11.2.2.1.3

Evaluate $\sin (52)$ .

$\sin (B) = \frac{117 \cdot 0.78801075}{89}$

Step 11.2.2.1.4

Multiply $117$ by $0.78801075$ .

$\sin (B) = \frac{92.19725817}{89}$

Step 11.2.2.1.5

Divide $92.19725817$ by $89$ .

$\sin (B) = 1.03592424$

Step 11.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.03592424$ 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 $B$ .

$\frac{\sin (B)}{234} = \frac{\sin (52)}{178}$

Step 15

Solve the equation for $B$ .

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Step 15.1

Multiply both sides of the equation by $234$ .

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

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 $234$ .

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Step 15.2.1.1.1

Cancel the common factor.

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

Step 15.2.1.1.2

Rewrite the expression.

$\sin (B) = 234 \frac{\sin (52)}{178}$

Step 15.2.2

Simplify the right side.

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Step 15.2.2.1

Simplify $234 \frac{\sin (52)}{178}$ .

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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 $234$ .

$\sin (B) = 2 (117) \frac{\sin (52)}{178}$

Step 15.2.2.1.1.2

Factor $2$ out of $178$ .

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 15.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 15.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 117 \frac{\sin (52)}{89}$

Step 15.2.2.1.2

Combine $117$ and $\frac{\sin (52)}{89}$ .

$\sin (B) = \frac{117 \sin (52)}{89}$

Step 15.2.2.1.3

Evaluate $\sin (52)$ .

$\sin (B) = \frac{117 \cdot 0.78801075}{89}$

Step 15.2.2.1.4

Multiply $117$ by $0.78801075$ .

$\sin (B) = \frac{92.19725817}{89}$

Step 15.2.2.1.5

Divide $92.19725817$ by $89$ .

$\sin (B) = 1.03592424$

Step 15.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.03592424$ 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 $B$ .

$\frac{\sin (B)}{234} = \frac{\sin (52)}{178}$

Step 19

Solve the equation for $B$ .

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Step 19.1

Multiply both sides of the equation by $234$ .

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

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 $234$ .

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Step 19.2.1.1.1

Cancel the common factor.

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

Step 19.2.1.1.2

Rewrite the expression.

$\sin (B) = 234 \frac{\sin (52)}{178}$

Step 19.2.2

Simplify the right side.

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Step 19.2.2.1

Simplify $234 \frac{\sin (52)}{178}$ .

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Step 19.2.2.1.1

Cancel the common factor of $2$ .

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Step 19.2.2.1.1.1

Factor $2$ out of $234$ .

$\sin (B) = 2 (117) \frac{\sin (52)}{178}$

Step 19.2.2.1.1.2

Factor $2$ out of $178$ .

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 19.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 19.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 117 \frac{\sin (52)}{89}$

Step 19.2.2.1.2

Combine $117$ and $\frac{\sin (52)}{89}$ .

$\sin (B) = \frac{117 \sin (52)}{89}$

Step 19.2.2.1.3

Evaluate $\sin (52)$ .

$\sin (B) = \frac{117 \cdot 0.78801075}{89}$

Step 19.2.2.1.4

Multiply $117$ by $0.78801075$ .

$\sin (B) = \frac{92.19725817}{89}$

Step 19.2.2.1.5

Divide $92.19725817$ by $89$ .

$\sin (B) = 1.03592424$

Step 19.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.03592424$ 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 $B$ .

$\frac{\sin (B)}{234} = \frac{\sin (52)}{178}$

Step 23

Solve the equation for $B$ .

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Step 23.1

Multiply both sides of the equation by $234$ .

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

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 $234$ .

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Step 23.2.1.1.1

Cancel the common factor.

$234 \frac{\sin (B)}{234} = 234 \frac{\sin (52)}{178}$

Step 23.2.1.1.2

Rewrite the expression.

$\sin (B) = 234 \frac{\sin (52)}{178}$

Step 23.2.2

Simplify the right side.

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Step 23.2.2.1

Simplify $234 \frac{\sin (52)}{178}$ .

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Step 23.2.2.1.1

Cancel the common factor of $2$ .

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Step 23.2.2.1.1.1

Factor $2$ out of $234$ .

$\sin (B) = 2 (117) \frac{\sin (52)}{178}$

Step 23.2.2.1.1.2

Factor $2$ out of $178$ .

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 23.2.2.1.1.3

Cancel the common factor.

$\sin (B) = 2 \cdot 117 \frac{\sin (52)}{2 \cdot 89}$

Step 23.2.2.1.1.4

Rewrite the expression.

$\sin (B) = 117 \frac{\sin (52)}{89}$

Step 23.2.2.1.2

Combine $117$ and $\frac{\sin (52)}{89}$ .

$\sin (B) = \frac{117 \sin (52)}{89}$

Step 23.2.2.1.3

Evaluate $\sin (52)$ .

$\sin (B) = \frac{117 \cdot 0.78801075}{89}$

Step 23.2.2.1.4

Multiply $117$ by $0.78801075$ .

$\sin (B) = \frac{92.19725817}{89}$

Step 23.2.2.1.5

Divide $92.19725817$ by $89$ .

$\sin (B) = 1.03592424$

Step 23.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.03592424$ 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