Trigonometry Examples

Solve for x 4cos(x)^2-4 square root of 2cos(x)+2=0
Step 1
Substitute for .
Step 2
Factor out of .
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Step 2.1
Factor out of .
Step 2.2
Factor out of .
Step 2.3
Factor out of .
Step 2.4
Factor out of .
Step 2.5
Factor out of .
Step 3
Divide each term in by and simplify.
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Step 3.1
Divide each term in by .
Step 3.2
Simplify the left side.
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Step 3.2.1
Cancel the common factor of .
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Step 3.2.1.1
Cancel the common factor.
Step 3.2.1.2
Divide by .
Step 3.3
Simplify the right side.
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Step 3.3.1
Divide by .
Step 4
Use the quadratic formula to find the solutions.
Step 5
Substitute the values , , and into the quadratic formula and solve for .
Step 6
Simplify.
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Step 6.1
Simplify the numerator.
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Step 6.1.1
Apply the product rule to .
Step 6.1.2
Raise to the power of .
Step 6.1.3
Rewrite as .
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Step 6.1.3.1
Use to rewrite as .
Step 6.1.3.2
Apply the power rule and multiply exponents, .
Step 6.1.3.3
Combine and .
Step 6.1.3.4
Cancel the common factor of .
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Step 6.1.3.4.1
Cancel the common factor.
Step 6.1.3.4.2
Rewrite the expression.
Step 6.1.3.5
Evaluate the exponent.
Step 6.1.4
Multiply by .
Step 6.1.5
Multiply .
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Step 6.1.5.1
Multiply by .
Step 6.1.5.2
Multiply by .
Step 6.1.6
Subtract from .
Step 6.1.7
Rewrite as .
Step 6.1.8
Pull terms out from under the radical, assuming positive real numbers.
Step 6.1.9
plus or minus is .
Step 6.2
Multiply by .
Step 6.3
Cancel the common factor of and .
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Step 6.3.1
Factor out of .
Step 6.3.2
Cancel the common factors.
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Step 6.3.2.1
Factor out of .
Step 6.3.2.2
Cancel the common factor.
Step 6.3.2.3
Rewrite the expression.
Step 7
Substitute for .
Step 8
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 9
Simplify the right side.
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Step 9.1
The exact value of is .
Step 10
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 11
Simplify .
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Step 11.1
To write as a fraction with a common denominator, multiply by .
Step 11.2
Combine fractions.
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Step 11.2.1
Combine and .
Step 11.2.2
Combine the numerators over the common denominator.
Step 11.3
Simplify the numerator.
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Step 11.3.1
Multiply by .
Step 11.3.2
Subtract from .
Step 12
Find the period of .
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Step 12.1
The period of the function can be calculated using .
Step 12.2
Replace with in the formula for period.
Step 12.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 12.4
Divide by .
Step 13
The period of the function is so values will repeat every radians in both directions.
, for any integer