Calculus Examples

Solve for x 2cos(2x)=0
Step 1
Divide each term in by and simplify.
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Step 1.1
Divide each term in by .
Step 1.2
Simplify the left side.
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Step 1.2.1
Cancel the common factor of .
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Step 1.2.1.1
Cancel the common factor.
Step 1.2.1.2
Divide by .
Step 1.3
Simplify the right side.
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Step 1.3.1
Divide by .
Step 2
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 3
Simplify the right side.
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Step 3.1
The exact value of is .
Step 4
Divide each term in by and simplify.
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Step 4.1
Divide each term in by .
Step 4.2
Simplify the left side.
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Step 4.2.1
Cancel the common factor of .
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Step 4.2.1.1
Cancel the common factor.
Step 4.2.1.2
Divide by .
Step 4.3
Simplify the right side.
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Step 4.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 4.3.2
Multiply .
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Step 4.3.2.1
Multiply by .
Step 4.3.2.2
Multiply by .
Step 5
The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the fourth quadrant.
Step 6
Solve for .
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Step 6.1
Simplify.
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Step 6.1.1
To write as a fraction with a common denominator, multiply by .
Step 6.1.2
Combine and .
Step 6.1.3
Combine the numerators over the common denominator.
Step 6.1.4
Multiply by .
Step 6.1.5
Subtract from .
Step 6.2
Divide each term in by and simplify.
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Step 6.2.1
Divide each term in by .
Step 6.2.2
Simplify the left side.
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Step 6.2.2.1
Cancel the common factor of .
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Step 6.2.2.1.1
Cancel the common factor.
Step 6.2.2.1.2
Divide by .
Step 6.2.3
Simplify the right side.
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Step 6.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.2.3.2
Multiply .
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Step 6.2.3.2.1
Multiply by .
Step 6.2.3.2.2
Multiply by .
Step 7
Find the period of .
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Step 7.1
The period of the function can be calculated using .
Step 7.2
Replace with in the formula for period.
Step 7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 7.4
Cancel the common factor of .
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Step 7.4.1
Cancel the common factor.
Step 7.4.2
Divide by .
Step 8
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 9
Consolidate the answers.
, for any integer