cos^2 (2x) = 1/4

Precalculus Examples

Step 1

Take the specified root of both sides of the equation lớn eliminate the exponent on the left side.

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Step 2

Simplify .

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Step 2.1

Step 2.2

Step 2.3

Simplify the denominator.

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Step 2.3.1

Step 2.3.2

Pull terms out from under the radical, assuming positive real numbers.

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.1

First, use the positive value of the lớn find the first solution.

Step 3.2

Next, use the negative value of the lớn find the second solution.

Step 3.3

The complete solution is the result of both the positive and negative portions of the solution.

Step 4

Set up each of the solutions lớn solve for .

Step 5

Solve for in .

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Step 5.1

Take the inverse cosine of both sides of the equation lớn extract from inside the cosine.

Step 5.2

Simplify the right side.

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Step 5.3

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from lớn find the solution in the fourth quadrant.

Step 5.4

Simplify .

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Step 5.4.1

To write as a fraction with a common denominator, multiply by .

Step 5.4.2

Combine fractions.

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Step 5.4.2.1

Step 5.4.2.2

Combine the numerators over the common denominator.

Step 5.4.3

Simplify the numerator.

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Step 5.5

Find the period of .

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Step 5.5.1

The period of the function can be calculated using .

Step 5.5.2

Replace with in the formula for period.

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Step 5.5.3

The absolute value is the distance between a number and zero. The distance between and is .

Step 5.5.4

Step 5.6

The period of the function is so sánh values will repeat every radians in both directions.

, for any integer

Step 6

Solve for in .

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Step 6.1

Take the inverse cosine of both sides of the equation lớn extract from inside the cosine.

Step 6.2

Simplify the right side.

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Step 6.3

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from lớn find the solution in the third quadrant.

Step 6.4

Simplify .

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Step 6.4.1

To write as a fraction with a common denominator, multiply by .

Step 6.4.2

Combine fractions.

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Step 6.4.2.1

Step 6.4.2.2

Combine the numerators over the common denominator.

Step 6.4.3

Simplify the numerator.

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Step 6.5

Find the period of .

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Step 6.5.1

The period of the function can be calculated using .

Step 6.5.2

Replace with in the formula for period.

Step 6.5.3

The absolute value is the distance between a number and zero. The distance between and is .

Step 6.5.4

Step 6.6

The period of the function is so sánh values will repeat every radians in both directions.

, for any integer

Step 7

List all of the solutions.

, for any integer

Step 8

Consolidate the solutions.

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Step 8.1

Consolidate and lớn .

, for any integer

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Step 8.2

Consolidate and lớn .

, for any integer