Trigonometry Examples

Solve for x 2cos(1/2x)- square root of 2=0
Step 1
Add to both sides of the equation.
Step 2
Divide each term in by and simplify.
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Step 2.1
Divide each term in by .
Step 2.2
Simplify the left side.
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Step 2.2.1
Cancel the common factor of .
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Step 2.2.1.1
Cancel the common factor.
Step 2.2.1.2
Divide by .
Step 3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 4
Simplify the left side.
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Step 4.1
Combine and .
Step 5
Simplify the right side.
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Step 5.1
The exact value of is .
Step 6
Multiply both sides of the equation by .
Step 7
Simplify both sides of the equation.
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Step 7.1
Simplify the left side.
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Step 7.1.1
Cancel the common factor of .
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Step 7.1.1.1
Cancel the common factor.
Step 7.1.1.2
Rewrite the expression.
Step 7.2
Simplify the right side.
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Step 7.2.1
Cancel the common factor of .
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Step 7.2.1.1
Factor out of .
Step 7.2.1.2
Cancel the common factor.
Step 7.2.1.3
Rewrite the expression.
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
Solve for .
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Step 9.1
Multiply both sides of the equation by .
Step 9.2
Simplify both sides of the equation.
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Step 9.2.1
Simplify the left side.
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Step 9.2.1.1
Cancel the common factor of .
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Step 9.2.1.1.1
Cancel the common factor.
Step 9.2.1.1.2
Rewrite the expression.
Step 9.2.2
Simplify the right side.
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Step 9.2.2.1
Simplify .
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Step 9.2.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 9.2.2.1.2
Simplify terms.
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Step 9.2.2.1.2.1
Combine and .
Step 9.2.2.1.2.2
Combine the numerators over the common denominator.
Step 9.2.2.1.2.3
Cancel the common factor of .
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Step 9.2.2.1.2.3.1
Factor out of .
Step 9.2.2.1.2.3.2
Cancel the common factor.
Step 9.2.2.1.2.3.3
Rewrite the expression.
Step 9.2.2.1.3
Simplify the numerator.
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Step 9.2.2.1.3.1
Multiply by .
Step 9.2.2.1.3.2
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
is approximately which is positive so remove the absolute value
Step 10.4
Multiply the numerator by the reciprocal of the denominator.
Step 10.5
Multiply by .
Step 11
The period of the function is so values will repeat every radians in both directions.
, for any integer