Algebra Examples

Find the Roots (Zeros) 3cos(x+pi/6)
Step 1
Set equal to .
Step 2
Solve for .
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Step 2.1
Divide each term in by and simplify.
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Step 2.1.1
Divide each term in by .
Step 2.1.2
Simplify the left side.
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Step 2.1.2.1
Cancel the common factor of .
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Step 2.1.2.1.1
Cancel the common factor.
Step 2.1.2.1.2
Divide by .
Step 2.1.3
Simplify the right side.
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Step 2.1.3.1
Divide by .
Step 2.2
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2.3
Simplify the right side.
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Step 2.3.1
The exact value of is .
Step 2.4
Move all terms not containing to the right side of the equation.
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Step 2.4.1
Subtract from both sides of the equation.
Step 2.4.2
To write as a fraction with a common denominator, multiply by .
Step 2.4.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 2.4.3.1
Multiply by .
Step 2.4.3.2
Multiply by .
Step 2.4.4
Combine the numerators over the common denominator.
Step 2.4.5
Simplify the numerator.
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Step 2.4.5.1
Move to the left of .
Step 2.4.5.2
Subtract from .
Step 2.4.6
Cancel the common factor of and .
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Step 2.4.6.1
Factor out of .
Step 2.4.6.2
Cancel the common factors.
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Step 2.4.6.2.1
Factor out of .
Step 2.4.6.2.2
Cancel the common factor.
Step 2.4.6.2.3
Rewrite the expression.
Step 2.5
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 2.6
Solve for .
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Step 2.6.1
Simplify .
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Step 2.6.1.1
To write as a fraction with a common denominator, multiply by .
Step 2.6.1.2
Combine fractions.
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Step 2.6.1.2.1
Combine and .
Step 2.6.1.2.2
Combine the numerators over the common denominator.
Step 2.6.1.3
Simplify the numerator.
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Step 2.6.1.3.1
Multiply by .
Step 2.6.1.3.2
Subtract from .
Step 2.6.2
Move all terms not containing to the right side of the equation.
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Step 2.6.2.1
Subtract from both sides of the equation.
Step 2.6.2.2
To write as a fraction with a common denominator, multiply by .
Step 2.6.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 2.6.2.3.1
Multiply by .
Step 2.6.2.3.2
Multiply by .
Step 2.6.2.4
Combine the numerators over the common denominator.
Step 2.6.2.5
Simplify the numerator.
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Step 2.6.2.5.1
Multiply by .
Step 2.6.2.5.2
Subtract from .
Step 2.6.2.6
Cancel the common factor of and .
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Step 2.6.2.6.1
Factor out of .
Step 2.6.2.6.2
Cancel the common factors.
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Step 2.6.2.6.2.1
Factor out of .
Step 2.6.2.6.2.2
Cancel the common factor.
Step 2.6.2.6.2.3
Rewrite the expression.
Step 2.7
Find the period of .
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Step 2.7.1
The period of the function can be calculated using .
Step 2.7.2
Replace with in the formula for period.
Step 2.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 2.7.4
Divide by .
Step 2.8
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 2.9
Consolidate the answers.
, for any integer
, for any integer
Step 3