# Solve the Triangle C=30 , a=32 , c=16

Solve the Triangle C=30 , a=32 , c=16
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The Law of Sines produces an ambiguous angle result. This means that there are angles that will correctly solve the equation. For the first triangle, use the first possible angle value.
Solve for the first 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 .
Solve the equation for .
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 .
The sum of all the angles in a triangle is degrees.
Solve the equation for .
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 .
Solve the equation for .
Factor each term.
The exact value of is .
Multiply the numerator by the reciprocal of the denominator.
Multiply and .
The exact value of is .
Set up the rational expression with the same denominator over the entire equation.
Multiply each term by a factor of that will equate all the denominators. In this case, all terms need a denominator of . The expression needs to be multiplied by to make the denominator . The expression needs to be multiplied by to make the denominator .
Multiply the expression by a factor of to create the least common denominator (LCD) of .
Move to the left of .
Multiply the expression by a factor of to create the least common denominator (LCD) of .
Multiply by .
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Rewrite the equation as .
For the second triangle, use the second possible angle value.
Solve for the second 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 .
Solve the equation for .
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 .
The sum of all the angles in a triangle is degrees.
Solve the equation for .
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 .
Solve the equation for .
Factor each term.
The exact value of is .
Multiply the numerator by the reciprocal of the denominator.
Multiply and .
The exact value of is .
Set up the rational expression with the same denominator over the entire equation.
Multiply each term by a factor of that will equate all the denominators. In this case, all terms need a denominator of . The expression needs to be multiplied by to make the denominator . The expression needs to be multiplied by to make the denominator .
Multiply the expression by a factor of to create the least common denominator (LCD) of .
Move to the left of .
Multiply the expression by a factor of to create the least common denominator (LCD) of .
Multiply by .
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Rewrite the equation as .
These are the results for all angles and sides for the given triangle.
First Triangle Combination:
Second Triangle Combination:
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