(sin x/2 + cos x/2) ^ 2 + sqrt(3) * cos x = 2

Trigonometry Examples

Solve for x sin(x)^2-cos(x)^2=( square root of 3)/2

Step 1

Bạn đang xem: (sin x/2 + cos x/2) ^ 2 + sqrt(3) * cos x = 2

Subtract from both sides of the equation.

Step 2

Replace the with based on the identity.

Step 5

Move all terms not containing đồ sộ the right side of the equation.

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Step 5.1

Subtract from both sides of the equation.

Step 5.2

Add đồ sộ both sides of the equation.

Step 6

Divide each term in by and simplify.

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Step 6.1

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of .

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Step 6.2.1.1

Cancel the common factor.

Step 6.2.1.2

Step 6.3

Simplify the right side.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Dividing two negative values results in a positive value.

Step 6.3.1.2

Multiply the numerator by the reciprocal of the denominator.

Step 6.3.1.3

Move the negative in front of the fraction.

Step 6.3.1.4

Multiply .

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Step 6.3.1.4.1

Step 6.3.1.4.2

Step 7

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

Step 8

Simplify .

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Step 8.1

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

Step 8.2

Write each expression with a common denominator of , by multiplying each by an appropriate factor of .

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Step 8.3

Combine the numerators over the common denominator.

Step 8.4

Step 8.5

Simplify the denominator.

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Step 8.5.1

Step 8.5.2

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

Step 9

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

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Step 9.1

First, use the positive value of the đồ sộ find the first solution.

Step 9.2

Next, use the negative value of the đồ sộ find the second solution.

Step 9.3

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

Step 10

Set up each of the solutions đồ sộ solve for .

Step 11

Solve for in .

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Step 11.1

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

Step 11.2

Simplify the right side.

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Step 11.3

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

Step 11.4

Simplify .

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Step 11.4.1

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

Step 11.4.2

Combine fractions.

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Step 11.4.2.1

Step 11.4.2.2

Combine the numerators over the common denominator.

Step 11.4.3

Simplify the numerator.

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Step 11.4.3.1

Step 11.4.3.2

Step 11.5

Find the period of .

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Step 11.5.1

The period of the function can be calculated using .

Step 11.5.2

Replace with in the formula for period.

Step 11.5.3

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

Step 11.5.4

Step 11.6

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

, for any integer

Step 12

Solve for in .

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Step 12.1

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

Step 12.2

Simplify the right side.

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Step 12.3

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

Step 12.4

Simplify .

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Step 12.4.1

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

Step 12.4.2

Combine fractions.

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Step 12.4.2.1

Step 12.4.2.2

Combine the numerators over the common denominator.

Step 12.4.3

Simplify the numerator.

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Step 12.4.3.1

Step 12.4.3.2

Step 12.5

Find the period of .

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Step 12.5.1

The period of the function can be calculated using .

Step 12.5.2

Replace with in the formula for period.

Step 12.5.3

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

Step 12.5.4

Step 12.6

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

, for any integer

Step 13

List all of the solutions.

, for any integer

Step 14

Consolidate the solutions.

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Step 14.1

Consolidate and đồ sộ .

, for any integer

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Step 14.2

Consolidate and đồ sộ .

, for any integer