Trigonometry Examples

Solve the Triangle A=34 , a=8 , b=13
, ,
Step 1
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.
Step 2
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.
Step 3
Substitute the known values into the law of sines to find .
Step 4
Solve the equation for .
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Step 4.1
Multiply both sides of the equation by .
Step 4.2
Simplify both sides of the equation.
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Step 4.2.1
Simplify the left side.
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Step 4.2.1.1
Cancel the common factor of .
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Step 4.2.1.1.1
Cancel the common factor.
Step 4.2.1.1.2
Rewrite the expression.
Step 4.2.2
Simplify the right side.
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Step 4.2.2.1
Simplify .
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Step 4.2.2.1.1
Evaluate .
Step 4.2.2.1.2
Divide by .
Step 4.2.2.1.3
Multiply by .
Step 4.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 4.4
Simplify the right side.
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Step 4.4.1
Evaluate .
Step 4.5
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.
Step 4.6
Subtract from .
Step 4.7
The solution to the equation .
Step 5
The sum of all the angles in a triangle is degrees.
Step 6
Solve the equation for .
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Step 6.1
Add and .
Step 6.2
Move all terms not containing to the right side of the equation.
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Step 6.2.1
Subtract from both sides of the equation.
Step 6.2.2
Subtract from .
Step 7
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.
Step 8
Substitute the known values into the law of sines to find .
Step 9
Solve the equation for .
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Step 9.1
Factor each term.
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Step 9.1.1
Evaluate .
Step 9.1.2
Evaluate .
Step 9.1.3
Divide by .
Step 9.2
Find the LCD of the terms in the equation.
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Step 9.2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 9.2.2
The LCM of one and any expression is the expression.
Step 9.3
Multiply each term in by to eliminate the fractions.
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Step 9.3.1
Multiply each term in by .
Step 9.3.2
Simplify the left side.
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Step 9.3.2.1
Cancel the common factor of .
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Step 9.3.2.1.1
Cancel the common factor.
Step 9.3.2.1.2
Rewrite the expression.
Step 9.4
Solve the equation.
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Step 9.4.1
Rewrite the equation as .
Step 9.4.2
Divide each term in by and simplify.
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Step 9.4.2.1
Divide each term in by .
Step 9.4.2.2
Simplify the left side.
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Step 9.4.2.2.1
Cancel the common factor of .
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Step 9.4.2.2.1.1
Cancel the common factor.
Step 9.4.2.2.1.2
Divide by .
Step 9.4.2.3
Simplify the right side.
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Step 9.4.2.3.1
Divide by .
Step 10
For the second triangle, use the second possible angle value.
Solve for the second triangle.
Step 11
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.
Step 12
Substitute the known values into the law of sines to find .
Step 13
Solve the equation for .
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Step 13.1
Multiply both sides of the equation by .
Step 13.2
Simplify both sides of the equation.
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Step 13.2.1
Simplify the left side.
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Step 13.2.1.1
Cancel the common factor of .
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Step 13.2.1.1.1
Cancel the common factor.
Step 13.2.1.1.2
Rewrite the expression.
Step 13.2.2
Simplify the right side.
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Step 13.2.2.1
Simplify .
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Step 13.2.2.1.1
Evaluate .
Step 13.2.2.1.2
Divide by .
Step 13.2.2.1.3
Multiply by .
Step 13.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 13.4
Simplify the right side.
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Step 13.4.1
Evaluate .
Step 13.5
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.
Step 13.6
Subtract from .
Step 13.7
The solution to the equation .
Step 14
The sum of all the angles in a triangle is degrees.
Step 15
Solve the equation for .
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Step 15.1
Add and .
Step 15.2
Move all terms not containing to the right side of the equation.
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Step 15.2.1
Subtract from both sides of the equation.
Step 15.2.2
Subtract from .
Step 16
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.
Step 17
Substitute the known values into the law of sines to find .
Step 18
Solve the equation for .
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Step 18.1
Factor each term.
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Step 18.1.1
Evaluate .
Step 18.1.2
Evaluate .
Step 18.1.3
Divide by .
Step 18.2
Find the LCD of the terms in the equation.
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Step 18.2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 18.2.2
The LCM of one and any expression is the expression.
Step 18.3
Multiply each term in by to eliminate the fractions.
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Step 18.3.1
Multiply each term in by .
Step 18.3.2
Simplify the left side.
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Step 18.3.2.1
Cancel the common factor of .
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Step 18.3.2.1.1
Cancel the common factor.
Step 18.3.2.1.2
Rewrite the expression.
Step 18.4
Solve the equation.
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Step 18.4.1
Rewrite the equation as .
Step 18.4.2
Divide each term in by and simplify.
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Step 18.4.2.1
Divide each term in by .
Step 18.4.2.2
Simplify the left side.
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Step 18.4.2.2.1
Cancel the common factor of .
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Step 18.4.2.2.1.1
Cancel the common factor.
Step 18.4.2.2.1.2
Divide by .
Step 18.4.2.3
Simplify the right side.
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Step 18.4.2.3.1
Divide by .
Step 19
These are the results for all angles and sides for the given triangle.
First Triangle Combination:
Second Triangle Combination: