Trigonometry Examples

Solve for x 4cos(x)^2-4cos(x)+1=0
Step 1
Substitute for .
Step 2
Factor using the perfect square rule.
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Step 2.1
Rewrite as .
Step 2.2
Rewrite as .
Step 2.3
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 2.4
Rewrite the polynomial.
Step 2.5
Factor using the perfect square trinomial rule , where and .
Step 3
Set the equal to .
Step 4
Solve for .
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Step 4.1
Add to both sides of the equation.
Step 4.2
Divide each term in by and simplify.
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Step 4.2.1
Divide each term in by .
Step 4.2.2
Simplify the left side.
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Step 4.2.2.1
Cancel the common factor of .
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Step 4.2.2.1.1
Cancel the common factor.
Step 4.2.2.1.2
Divide by .
Step 5
Substitute for .
Step 6
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 7
Simplify the right side.
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Step 7.1
The exact value of is .
Step 8
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 9
Simplify .
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Step 9.1
To write as a fraction with a common denominator, multiply by .
Step 9.2
Combine fractions.
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Step 9.2.1
Combine and .
Step 9.2.2
Combine the numerators over the common denominator.
Step 9.3
Simplify the numerator.
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Step 9.3.1
Multiply by .
Step 9.3.2
Subtract from .
Step 10
Find the period of .
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Step 10.1
The period of the function can be calculated using .
Step 10.2
Replace with in the formula for period.
Step 10.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.4
Divide by .
Step 11
The period of the function is so values will repeat every radians in both directions.
, for any integer