Trigonometry Examples

Solve the Triangle tri{23}{}{}{105}{14}{}
Step 1
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 2
Substitute the known values into the law of sines to find .
Step 3
Solve the equation for .
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Step 3.1
Multiply both sides of the equation by .
Step 3.2
Simplify both sides of the equation.
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Step 3.2.1
Simplify the left side.
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Step 3.2.1.1
Cancel the common factor of .
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Step 3.2.1.1.1
Cancel the common factor.
Step 3.2.1.1.2
Rewrite the expression.
Step 3.2.2
Simplify the right side.
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Step 3.2.2.1
Simplify .
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Step 3.2.2.1.1
The exact value of is .
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Step 3.2.2.1.1.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 3.2.2.1.1.2
Split into two angles where the values of the six trigonometric functions are known.
Step 3.2.2.1.1.3
Apply the sum of angles identity.
Step 3.2.2.1.1.4
The exact value of is .
Step 3.2.2.1.1.5
The exact value of is .
Step 3.2.2.1.1.6
The exact value of is .
Step 3.2.2.1.1.7
The exact value of is .
Step 3.2.2.1.1.8
Simplify .
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Step 3.2.2.1.1.8.1
Simplify each term.
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Step 3.2.2.1.1.8.1.1
Multiply .
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Step 3.2.2.1.1.8.1.1.1
Multiply by .
Step 3.2.2.1.1.8.1.1.2
Multiply by .
Step 3.2.2.1.1.8.1.2
Multiply .
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Step 3.2.2.1.1.8.1.2.1
Multiply by .
Step 3.2.2.1.1.8.1.2.2
Combine using the product rule for radicals.
Step 3.2.2.1.1.8.1.2.3
Multiply by .
Step 3.2.2.1.1.8.1.2.4
Multiply by .
Step 3.2.2.1.1.8.2
Combine the numerators over the common denominator.
Step 3.2.2.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 3.2.2.1.3
Multiply .
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Step 3.2.2.1.3.1
Multiply by .
Step 3.2.2.1.3.2
Multiply by .
Step 3.2.2.1.4
Cancel the common factor of .
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Step 3.2.2.1.4.1
Factor out of .
Step 3.2.2.1.4.2
Factor out of .
Step 3.2.2.1.4.3
Cancel the common factor.
Step 3.2.2.1.4.4
Rewrite the expression.
Step 3.2.2.1.5
Combine and .
Step 3.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 3.4
Simplify the right side.
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Step 3.4.1
Evaluate .
Step 3.5
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 3.6
Subtract from .
Step 3.7
The solution to the equation .
Step 3.8
Exclude the invalid angle.
Step 4
The sum of all the angles in a triangle is degrees.
Step 5
Solve the equation for .
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Step 5.1
Add and .
Step 5.2
Move all terms not containing to the right side of the equation.
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Step 5.2.1
Subtract from both sides of the equation.
Step 5.2.2
Subtract from .
Step 6
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 7
Substitute the known values into the law of sines to find .
Step 8
Solve the equation for .
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Step 8.1
Factor each term.
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Step 8.1.1
Evaluate .
Step 8.1.2
Evaluate .
Step 8.1.3
Divide by .
Step 8.2
Find the LCD of the terms in the equation.
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Step 8.2.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 8.2.2
The LCM of one and any expression is the expression.
Step 8.3
Multiply each term in by to eliminate the fractions.
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Step 8.3.1
Multiply each term in by .
Step 8.3.2
Simplify the left side.
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Step 8.3.2.1
Cancel the common factor of .
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Step 8.3.2.1.1
Cancel the common factor.
Step 8.3.2.1.2
Rewrite the expression.
Step 8.4
Solve the equation.
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Step 8.4.1
Rewrite the equation as .
Step 8.4.2
Divide each term in by and simplify.
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Step 8.4.2.1
Divide each term in by .
Step 8.4.2.2
Simplify the left side.
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Step 8.4.2.2.1
Cancel the common factor of .
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Step 8.4.2.2.1.1
Cancel the common factor.
Step 8.4.2.2.1.2
Divide by .
Step 8.4.2.3
Simplify the right side.
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Step 8.4.2.3.1
Divide by .
Step 9
These are the results for all angles and sides for the given triangle.