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# Trigonometry Examples

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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 .

Simplify .

The exact value of is .

Multiply the numerator by the reciprocal of the denominator.

Multiply .

Multiply and .

Multiply by .

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

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

The exact value of is .

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 .

Exclude the solutions that do not make true.

No solution

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle

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 .

Simplify .

The exact value of is .

Multiply the numerator by the reciprocal of the denominator.

Multiply .

Multiply and .

Multiply by .

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

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

The exact value of is .

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 .

Exclude the solutions that do not make true.

No solution

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle

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 .

Simplify .

The exact value of is .

Multiply the numerator by the reciprocal of the denominator.

Multiply .

Multiply and .

Multiply by .

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

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

The exact value of is .

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 .

Exclude the solutions that do not make true.

No solution

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle

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

Simplify .

The exact value of is .

Multiply the numerator by the reciprocal of the denominator.

Multiply .

Multiply and .

Multiply by .

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

The exact value of is .

Subtract from .

The solution to the equation .

Exclude the solutions that do not make true.

No solution

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle

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

Simplify .

The exact value of is .

Multiply the numerator by the reciprocal of the denominator.

Multiply .

Multiply and .

Multiply by .

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

The exact value of is .

Subtract from .

The solution to the equation .

Exclude the solutions that do not make true.

No solution

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle

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

Simplify .

The exact value of is .

Multiply the numerator by the reciprocal of the denominator.

Multiply .

Multiply and .

Multiply by .

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

The exact value of is .

Subtract from .

The solution to the equation .

Exclude the solutions that do not make true.

No solution

No solution

There are not enough parameters given to solve the triangle.

Unknown triangle