Trigonometry Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 12 \\ c = \\ a = & 23 \end{matrix} & \begin{matrix} A = \\ B = & 44 \\ C = \end{matrix} \end{matrix}$

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 $A$ .

$\frac{\sin (A)}{23} = \frac{\sin (44)}{12}$

Step 3

Solve the equation for $A$ .

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Step 3.1

Multiply both sides of the equation by $23$ .

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

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 $23$ .

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Step 3.2.1.1.1

Cancel the common factor.

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

Step 3.2.1.1.2

Rewrite the expression.

$\sin (A) = 23 \frac{\sin (44)}{12}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $23 \frac{\sin (44)}{12}$ .

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Step 3.2.2.1.1

Evaluate $\sin (44)$ .

$\sin (A) = 23 (\frac{0.69465837}{12})$

Step 3.2.2.1.2

Divide $0.69465837$ by $12$ .

$\sin (A) = 23 \cdot 0.05788819$

Step 3.2.2.1.3

Multiply $23$ by $0.05788819$ .

$\sin (A) = 1.33142854$

Step 3.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.33142854$ 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 $A$ .

$\frac{\sin (A)}{23} = \frac{\sin (44)}{12}$

Step 7

Solve the equation for $A$ .

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Step 7.1

Multiply both sides of the equation by $23$ .

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

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 $23$ .

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Step 7.2.1.1.1

Cancel the common factor.

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

Step 7.2.1.1.2

Rewrite the expression.

$\sin (A) = 23 \frac{\sin (44)}{12}$

Step 7.2.2

Simplify the right side.

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Step 7.2.2.1

Simplify $23 \frac{\sin (44)}{12}$ .

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Step 7.2.2.1.1

Evaluate $\sin (44)$ .

$\sin (A) = 23 (\frac{0.69465837}{12})$

Step 7.2.2.1.2

Divide $0.69465837$ by $12$ .

$\sin (A) = 23 \cdot 0.05788819$

Step 7.2.2.1.3

Multiply $23$ by $0.05788819$ .

$\sin (A) = 1.33142854$

Step 7.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.33142854$ 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 $A$ .

$\frac{\sin (A)}{23} = \frac{\sin (44)}{12}$

Step 11

Solve the equation for $A$ .

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Step 11.1

Multiply both sides of the equation by $23$ .

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

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 $23$ .

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Step 11.2.1.1.1

Cancel the common factor.

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

Step 11.2.1.1.2

Rewrite the expression.

$\sin (A) = 23 \frac{\sin (44)}{12}$

Step 11.2.2

Simplify the right side.

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Step 11.2.2.1

Simplify $23 \frac{\sin (44)}{12}$ .

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Step 11.2.2.1.1

Evaluate $\sin (44)$ .

$\sin (A) = 23 (\frac{0.69465837}{12})$

Step 11.2.2.1.2

Divide $0.69465837$ by $12$ .

$\sin (A) = 23 \cdot 0.05788819$

Step 11.2.2.1.3

Multiply $23$ by $0.05788819$ .

$\sin (A) = 1.33142854$

Step 11.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.33142854$ 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 $A$ .

$\frac{\sin (A)}{23} = \frac{\sin (44)}{12}$

Step 15

Solve the equation for $A$ .

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Step 15.1

Multiply both sides of the equation by $23$ .

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

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 $23$ .

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Step 15.2.1.1.1

Cancel the common factor.

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

Step 15.2.1.1.2

Rewrite the expression.

$\sin (A) = 23 \frac{\sin (44)}{12}$

Step 15.2.2

Simplify the right side.

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Step 15.2.2.1

Simplify $23 \frac{\sin (44)}{12}$ .

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Step 15.2.2.1.1

Evaluate $\sin (44)$ .

$\sin (A) = 23 (\frac{0.69465837}{12})$

Step 15.2.2.1.2

Divide $0.69465837$ by $12$ .

$\sin (A) = 23 \cdot 0.05788819$

Step 15.2.2.1.3

Multiply $23$ by $0.05788819$ .

$\sin (A) = 1.33142854$

Step 15.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.33142854$ 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 $A$ .

$\frac{\sin (A)}{23} = \frac{\sin (44)}{12}$

Step 19

Solve the equation for $A$ .

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Step 19.1

Multiply both sides of the equation by $23$ .

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

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 $23$ .

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Step 19.2.1.1.1

Cancel the common factor.

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

Step 19.2.1.1.2

Rewrite the expression.

$\sin (A) = 23 \frac{\sin (44)}{12}$

Step 19.2.2

Simplify the right side.

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Step 19.2.2.1

Simplify $23 \frac{\sin (44)}{12}$ .

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Step 19.2.2.1.1

Evaluate $\sin (44)$ .

$\sin (A) = 23 (\frac{0.69465837}{12})$

Step 19.2.2.1.2

Divide $0.69465837$ by $12$ .

$\sin (A) = 23 \cdot 0.05788819$

Step 19.2.2.1.3

Multiply $23$ by $0.05788819$ .

$\sin (A) = 1.33142854$

Step 19.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.33142854$ 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 $A$ .

$\frac{\sin (A)}{23} = \frac{\sin (44)}{12}$

Step 23

Solve the equation for $A$ .

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Step 23.1

Multiply both sides of the equation by $23$ .

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

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 $23$ .

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Step 23.2.1.1.1

Cancel the common factor.

$23 \frac{\sin (A)}{23} = 23 \frac{\sin (44)}{12}$

Step 23.2.1.1.2

Rewrite the expression.

$\sin (A) = 23 \frac{\sin (44)}{12}$

Step 23.2.2

Simplify the right side.

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Step 23.2.2.1

Simplify $23 \frac{\sin (44)}{12}$ .

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Step 23.2.2.1.1

Evaluate $\sin (44)$ .

$\sin (A) = 23 (\frac{0.69465837}{12})$

Step 23.2.2.1.2

Divide $0.69465837$ by $12$ .

$\sin (A) = 23 \cdot 0.05788819$

Step 23.2.2.1.3

Multiply $23$ by $0.05788819$ .

$\sin (A) = 1.33142854$

Step 23.3

The range of sine is $- 1 \leq y \leq 1$ . Since $1.33142854$ 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