Solve the Triangle tri(5)()(13)()(12)(90 degrees )

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 5 \\ c = & 13 \\ a = & 12 \end{matrix} & \begin{matrix} A = \\ B = \\ C = & 90 \end{matrix} \end{matrix}$

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.

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

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

$\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$

Solve the equation for $A$ .

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The exact value of $\sin (90 °)$ is $1$ .

$\frac{\sin (A)}{12} = \frac{1}{13}$

Multiply both sides of the equation by $12$ .

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Simplify both sides of the equation.

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Cancel the common factor of $12$ .

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Cancel the common factor.

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Rewrite the expression.

$\sin (A) = 12 \cdot \frac{1}{13}$

Combine $12$ and $\frac{1}{13}$ .

$\sin (A) = \frac{12}{13}$

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

$A = \arcsin (\frac{12}{13})$

Evaluate $\arcsin (\frac{12}{13})$ .

$A = 67.38013505$

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.

$A = 180 - 67.38013505$

Subtract $67.38013505$ from $180$ .

$A = 112.61986494$

The solution to the equation $A = 67.38013505$ .

$A = 67.38013505, 112.61986494$

Exclude the solutions that do not make $\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$ true.

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle

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

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

$\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$

Solve the equation for $A$ .

Tap for more steps…

The exact value of $\sin (90 °)$ is $1$ .

$\frac{\sin (A)}{12} = \frac{1}{13}$

Multiply both sides of the equation by $12$ .

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Simplify both sides of the equation.

Tap for more steps…

Cancel the common factor of $12$ .

Tap for more steps…

Cancel the common factor.

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Rewrite the expression.

$\sin (A) = 12 \cdot \frac{1}{13}$

Combine $12$ and $\frac{1}{13}$ .

$\sin (A) = \frac{12}{13}$

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

$A = \arcsin (\frac{12}{13})$

Evaluate $\arcsin (\frac{12}{13})$ .

$A = 67.38013505$

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.

$A = 180 - 67.38013505$

Subtract $67.38013505$ from $180$ .

$A = 112.61986494$

The solution to the equation $A = 67.38013505$ .

$A = 67.38013505, 112.61986494$

Exclude the solutions that do not make $\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$ true.

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle

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

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

$\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$

Solve the equation for $A$ .

Tap for more steps…

The exact value of $\sin (90 °)$ is $1$ .

$\frac{\sin (A)}{12} = \frac{1}{13}$

Multiply both sides of the equation by $12$ .

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Simplify both sides of the equation.

Tap for more steps…

Cancel the common factor of $12$ .

Tap for more steps…

Cancel the common factor.

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Rewrite the expression.

$\sin (A) = 12 \cdot \frac{1}{13}$

Combine $12$ and $\frac{1}{13}$ .

$\sin (A) = \frac{12}{13}$

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

$A = \arcsin (\frac{12}{13})$

Evaluate $\arcsin (\frac{12}{13})$ .

$A = 67.38013505$

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.

$A = 180 - 67.38013505$

Subtract $67.38013505$ from $180$ .

$A = 112.61986494$

The solution to the equation $A = 67.38013505$ .

$A = 67.38013505, 112.61986494$

Exclude the solutions that do not make $\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$ true.

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle

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

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

$\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$

Solve the equation for $A$ .

Tap for more steps…

The exact value of $\sin (90 °)$ is $1$ .

$\frac{\sin (A)}{12} = \frac{1}{13}$

Multiply both sides of the equation by $12$ .

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Simplify both sides of the equation.

Tap for more steps…

Cancel the common factor of $12$ .

Tap for more steps…

Cancel the common factor.

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Rewrite the expression.

$\sin (A) = 12 \cdot \frac{1}{13}$

Combine $12$ and $\frac{1}{13}$ .

$\sin (A) = \frac{12}{13}$

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

$A = \arcsin (\frac{12}{13})$

Evaluate $\arcsin (\frac{12}{13})$ .

$A = 67.38013505$

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.

$A = 180 - 67.38013505$

Subtract $67.38013505$ from $180$ .

$A = 112.61986494$

The solution to the equation $A = 67.38013505$ .

$A = 67.38013505, 112.61986494$

Exclude the solutions that do not make $\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$ true.

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle

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

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

$\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$

Solve the equation for $A$ .

Tap for more steps…

The exact value of $\sin (90 °)$ is $1$ .

$\frac{\sin (A)}{12} = \frac{1}{13}$

Multiply both sides of the equation by $12$ .

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Simplify both sides of the equation.

Tap for more steps…

Cancel the common factor of $12$ .

Tap for more steps…

Cancel the common factor.

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Rewrite the expression.

$\sin (A) = 12 \cdot \frac{1}{13}$

Combine $12$ and $\frac{1}{13}$ .

$\sin (A) = \frac{12}{13}$

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

$A = \arcsin (\frac{12}{13})$

Evaluate $\arcsin (\frac{12}{13})$ .

$A = 67.38013505$

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.

$A = 180 - 67.38013505$

Subtract $67.38013505$ from $180$ .

$A = 112.61986494$

The solution to the equation $A = 67.38013505$ .

$A = 67.38013505, 112.61986494$

Exclude the solutions that do not make $\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$ true.

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle

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

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

$\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$

Solve the equation for $A$ .

Tap for more steps…

The exact value of $\sin (90 °)$ is $1$ .

$\frac{\sin (A)}{12} = \frac{1}{13}$

Multiply both sides of the equation by $12$ .

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Simplify both sides of the equation.

Tap for more steps…

Cancel the common factor of $12$ .

Tap for more steps…

Cancel the common factor.

$12 \cdot \frac{\sin (A)}{12} = 12 \cdot \frac{1}{13}$

Rewrite the expression.

$\sin (A) = 12 \cdot \frac{1}{13}$

Combine $12$ and $\frac{1}{13}$ .

$\sin (A) = \frac{12}{13}$

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

$A = \arcsin (\frac{12}{13})$

Evaluate $\arcsin (\frac{12}{13})$ .

$A = 67.38013505$

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.

$A = 180 - 67.38013505$

Subtract $67.38013505$ from $180$ .

$A = 112.61986494$

The solution to the equation $A = 67.38013505$ .

$A = 67.38013505, 112.61986494$

Exclude the solutions that do not make $\frac{\sin (A)}{12} = \frac{\sin (90 °)}{13}$ true.

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle

Solve the Triangle tri(5)()(13)()(12)(90 degrees )

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