, ,

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 .

Factor each term.

The exact value of is .

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

Split into two angles where the values of the six trigonometric functions are known.

Apply the sum of angles identity.

The exact value of is .

The exact value of is .

The exact value of is .

The exact value of is .

Simplify .

Simplify each term.

Multiply .

Multiply and .

Multiply by .

Multiply .

Multiply and .

Combine using the product rule for radicals.

Multiply by .

Multiply by .

Combine the numerators over the common denominator.

Multiply the numerator by the reciprocal of the denominator.

Multiply and .

Evaluate .

Divide by .

Solve for .

Multiply each term by and simplify.

Multiply each term in by .

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Rewrite the equation as .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Simplify .

Multiply the numerator by the reciprocal of the denominator.

Divide by .

Multiply .

Combine and .

Multiply by .

Divide by .

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 .

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 .

Factor each term.

Evaluate .

Evaluate .

Divide by .

Solve for .

Multiply each term by and simplify.

Multiply each term in by .

Cancel the common factor of .

Cancel the common factor.

Rewrite the expression.

Rewrite the equation as .

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Divide by .

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

Solve the Triangle A=35 , C=105 , a=21