Trigonometry Examples

Solve the Triangle A=45 , c=5 , a=4
, ,
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
The exact value of is .
Step 4.2.2.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 4.2.2.1.3
Multiply .
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Step 4.2.2.1.3.1
Multiply by .
Step 4.2.2.1.3.2
Multiply by .
Step 4.2.2.1.4
Combine and .
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
The exact value of is .
Step 9.1.3
Multiply the numerator by the reciprocal of the denominator.
Step 9.1.4
Multiply .
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Step 9.1.4.1
Multiply by .
Step 9.1.4.2
Multiply 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
Since contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
Step 9.2.3
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
Step 9.2.4
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 9.2.5
The prime factors for are .
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Step 9.2.5.1
has factors of and .
Step 9.2.5.2
has factors of and .
Step 9.2.6
Multiply .
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Step 9.2.6.1
Multiply by .
Step 9.2.6.2
Multiply by .
Step 9.2.7
The factor for is itself.
occurs time.
Step 9.2.8
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 9.2.9
The LCM for is the numeric part multiplied by the variable part.
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
Rewrite using the commutative property of multiplication.
Step 9.3.2.2
Multiply .
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Step 9.3.2.2.1
Combine and .
Step 9.3.2.2.2
Multiply by .
Step 9.3.2.3
Cancel the common factor of .
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Step 9.3.2.3.1
Cancel the common factor.
Step 9.3.2.3.2
Rewrite the expression.
Step 9.3.3
Simplify the right side.
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Step 9.3.3.1
Cancel the common factor of .
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Step 9.3.3.1.1
Factor out of .
Step 9.3.3.1.2
Cancel the common factor.
Step 9.3.3.1.3
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
Multiply by .
Step 9.4.2.3.2
Combine and simplify the denominator.
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Step 9.4.2.3.2.1
Multiply by .
Step 9.4.2.3.2.2
Raise to the power of .
Step 9.4.2.3.2.3
Raise to the power of .
Step 9.4.2.3.2.4
Use the power rule to combine exponents.
Step 9.4.2.3.2.5
Add and .
Step 9.4.2.3.2.6
Rewrite as .
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Step 9.4.2.3.2.6.1
Use to rewrite as .
Step 9.4.2.3.2.6.2
Apply the power rule and multiply exponents, .
Step 9.4.2.3.2.6.3
Combine and .
Step 9.4.2.3.2.6.4
Cancel the common factor of .
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Step 9.4.2.3.2.6.4.1
Cancel the common factor.
Step 9.4.2.3.2.6.4.2
Rewrite the expression.
Step 9.4.2.3.2.6.5
Evaluate the exponent.
Step 9.4.2.3.3
Multiply by .
Step 9.4.2.3.4
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
The exact value of is .
Step 13.2.2.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 13.2.2.1.3
Multiply .
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Step 13.2.2.1.3.1
Multiply by .
Step 13.2.2.1.3.2
Multiply by .
Step 13.2.2.1.4
Combine and .
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
The exact value of is .
Step 18.1.3
Multiply the numerator by the reciprocal of the denominator.
Step 18.1.4
Multiply .
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Step 18.1.4.1
Multiply by .
Step 18.1.4.2
Multiply 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
Since contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part then find LCM for the variable part .
Step 18.2.3
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
Step 18.2.4
The number is not a prime number because it only has one positive factor, which is itself.
Not prime
Step 18.2.5
The prime factors for are .
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Step 18.2.5.1
has factors of and .
Step 18.2.5.2
has factors of and .
Step 18.2.6
Multiply .
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Step 18.2.6.1
Multiply by .
Step 18.2.6.2
Multiply by .
Step 18.2.7
The factor for is itself.
occurs time.
Step 18.2.8
The LCM of is the result of multiplying all prime factors the greatest number of times they occur in either term.
Step 18.2.9
The LCM for is the numeric part multiplied by the variable part.
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
Rewrite using the commutative property of multiplication.
Step 18.3.2.2
Multiply .
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Step 18.3.2.2.1
Combine and .
Step 18.3.2.2.2
Multiply by .
Step 18.3.2.3
Cancel the common factor of .
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Step 18.3.2.3.1
Cancel the common factor.
Step 18.3.2.3.2
Rewrite the expression.
Step 18.3.3
Simplify the right side.
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Step 18.3.3.1
Cancel the common factor of .
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Step 18.3.3.1.1
Factor out of .
Step 18.3.3.1.2
Cancel the common factor.
Step 18.3.3.1.3
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
Multiply by .
Step 18.4.2.3.2
Combine and simplify the denominator.
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Step 18.4.2.3.2.1
Multiply by .
Step 18.4.2.3.2.2
Raise to the power of .
Step 18.4.2.3.2.3
Raise to the power of .
Step 18.4.2.3.2.4
Use the power rule to combine exponents.
Step 18.4.2.3.2.5
Add and .
Step 18.4.2.3.2.6
Rewrite as .
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Step 18.4.2.3.2.6.1
Use to rewrite as .
Step 18.4.2.3.2.6.2
Apply the power rule and multiply exponents, .
Step 18.4.2.3.2.6.3
Combine and .
Step 18.4.2.3.2.6.4
Cancel the common factor of .
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Step 18.4.2.3.2.6.4.1
Cancel the common factor.
Step 18.4.2.3.2.6.4.2
Rewrite the expression.
Step 18.4.2.3.2.6.5
Evaluate the exponent.
Step 18.4.2.3.3
Multiply by .
Step 18.4.2.3.4
Divide by .
Step 19
These are the results for all angles and sides for the given triangle.
First Triangle Combination:
Second Triangle Combination: