, ,

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

Solve the equation.

Substitute the known values into the equation.

Raise to the power of .

Raise to the power of .

Multiply by .

Multiply by .

Evaluate .

Multiply by .

Add and .

Subtract from .

Evaluate the root.

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.

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

Multiply both sides of the equation by .

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Simplify the right side.

Simplify .

Evaluate .

Divide by .

Multiply by .

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

Simplify the right side.

Evaluate .

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.

Subtract from .

The solution to the equation .

The sum of all the angles in a triangle is degrees.

Add and .

Move all terms not containing to the right side of the equation.

Subtract from both sides of the equation.

Subtract from .

These are the results for all angles and sides for the given triangle.

Solve the Triangle A=51 degrees , b=11 , c=7