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Trigonometry Examples
, ,
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
Step 4.1
Raise to the power of .
Step 4.2
Raise to the power of .
Step 4.3
Multiply .
Step 4.3.1
Multiply by .
Step 4.3.2
Multiply by .
Step 4.4
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.
Step 4.5
The exact value of is .
Step 4.6
Cancel the common factor of .
Step 4.6.1
Move the leading negative in into the numerator.
Step 4.6.2
Factor out of .
Step 4.6.3
Cancel the common factor.
Step 4.6.4
Rewrite the expression.
Step 4.7
Simplify the expression.
Step 4.7.1
Multiply by .
Step 4.7.2
Add and .
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
Step 7.1
Multiply both sides of the equation by .
Step 7.2
Simplify both sides of the equation.
Step 7.2.1
Simplify the left side.
Step 7.2.1.1
Cancel the common factor of .
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.
Step 7.2.2.1
Simplify .
Step 7.2.2.1.1
Simplify the numerator.
Step 7.2.2.1.1.1
Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.
Step 7.2.2.1.1.2
The exact value of is .
Step 7.2.2.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 7.2.2.1.3
Multiply by .
Step 7.2.2.1.4
Cancel the common factor of .
Step 7.2.2.1.4.1
Factor out of .
Step 7.2.2.1.4.2
Factor out of .
Step 7.2.2.1.4.3
Cancel the common factor.
Step 7.2.2.1.4.4
Rewrite the expression.
Step 7.2.2.1.5
Combine and .
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.
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
Step 9.1
Add and .
Step 9.2
Move all terms not containing to the right side of the equation.
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.