Trigonometry Examples

Solve the Triangle A=60 , b=6 , c=9
, ,
Step 1
Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.
Step 2
Solve the equation.
Step 3
Substitute the known values into the equation.
Step 4
Simplify the results.
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Step 4.1
Raise to the power of .
Step 4.2
Raise to the power of .
Step 4.3
Multiply .
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Step 4.3.1
Multiply by .
Step 4.3.2
Multiply by .
Step 4.4
The exact value of is .
Step 4.5
Cancel the common factor of .
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Step 4.5.1
Factor out of .
Step 4.5.2
Cancel the common factor.
Step 4.5.3
Rewrite the expression.
Step 4.6
Simplify by adding and subtracting.
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Step 4.6.1
Add and .
Step 4.6.2
Subtract from .
Step 4.7
Rewrite as .
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Step 4.7.1
Factor out of .
Step 4.7.2
Rewrite as .
Step 4.8
Pull terms out from under the radical.
Step 5
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 6
Substitute the known values into the law of sines to find .
Step 7
Solve the equation for .
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Step 7.1
Multiply both sides of the equation by .
Step 7.2
Simplify both sides of the equation.
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Step 7.2.1
Simplify the left side.
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Step 7.2.1.1
Cancel the common factor of .
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Step 7.2.1.1.1
Cancel the common factor.
Step 7.2.1.1.2
Rewrite the expression.
Step 7.2.2
Simplify the right side.
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Step 7.2.2.1
Simplify .
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Step 7.2.2.1.1
Cancel the common factor of .
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Step 7.2.2.1.1.1
Factor out of .
Step 7.2.2.1.1.2
Factor out of .
Step 7.2.2.1.1.3
Cancel the common factor.
Step 7.2.2.1.1.4
Rewrite the expression.
Step 7.2.2.1.2
Combine and .
Step 7.2.2.1.3
The exact value of is .
Step 7.2.2.1.4
Combine and .
Step 7.2.2.1.5
Reduce the expression by cancelling the common factors.
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Step 7.2.2.1.5.1
Reduce the expression by cancelling the common factors.
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Step 7.2.2.1.5.1.1
Cancel the common factor.
Step 7.2.2.1.5.1.2
Rewrite the expression.
Step 7.2.2.1.5.2
Divide by .
Step 7.2.2.1.6
Multiply by .
Step 7.2.2.1.7
Combine and simplify the denominator.
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Step 7.2.2.1.7.1
Multiply by .
Step 7.2.2.1.7.2
Raise to the power of .
Step 7.2.2.1.7.3
Raise to the power of .
Step 7.2.2.1.7.4
Use the power rule to combine exponents.
Step 7.2.2.1.7.5
Add and .
Step 7.2.2.1.7.6
Rewrite as .
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Step 7.2.2.1.7.6.1
Use to rewrite as .
Step 7.2.2.1.7.6.2
Apply the power rule and multiply exponents, .
Step 7.2.2.1.7.6.3
Combine and .
Step 7.2.2.1.7.6.4
Cancel the common factor of .
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Step 7.2.2.1.7.6.4.1
Cancel the common factor.
Step 7.2.2.1.7.6.4.2
Rewrite the expression.
Step 7.2.2.1.7.6.5
Evaluate the exponent.
Step 7.2.2.1.8
Simplify the numerator.
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Step 7.2.2.1.8.1
Combine using the product rule for radicals.
Step 7.2.2.1.8.2
Multiply by .
Step 7.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 7.4
Simplify the right side.
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Step 7.4.1
Evaluate .
Step 7.5
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 7.6
Subtract from .
Step 7.7
The solution to the equation .
Step 7.8
Exclude the invalid angle.
Step 8
The sum of all the angles in a triangle is degrees.
Step 9
Solve the equation for .
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Step 9.1
Add and .
Step 9.2
Move all terms not containing to the right side of the equation.
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Step 9.2.1
Subtract from both sides of the equation.
Step 9.2.2
Subtract from .
Step 10
These are the results for all angles and sides for the given triangle.