Trigonometry Examples

Solve for x sin(2x)=cos(2x)
Step 1
Divide each term in the equation by .
Step 2
Convert from to .
Step 3
Cancel the common factor of .
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Step 3.1
Cancel the common factor.
Step 3.2
Rewrite the expression.
Step 4
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 5
Simplify the right side.
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Step 5.1
The exact value of is .
Step 6
Divide each term in by and simplify.
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Step 6.1
Divide each term in by .
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
Divide by .
Step 6.3
Simplify the right side.
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Step 6.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 6.3.2
Multiply .
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Step 6.3.2.1
Multiply by .
Step 6.3.2.2
Multiply by .
Step 7
The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Step 8
Solve for .
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Step 8.1
Simplify.
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Step 8.1.1
To write as a fraction with a common denominator, multiply by .
Step 8.1.2
Combine and .
Step 8.1.3
Combine the numerators over the common denominator.
Step 8.1.4
Add and .
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Step 8.1.4.1
Reorder and .
Step 8.1.4.2
Add and .
Step 8.2
Divide each term in by and simplify.
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Step 8.2.1
Divide each term in by .
Step 8.2.2
Simplify the left side.
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Step 8.2.2.1
Cancel the common factor of .
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Step 8.2.2.1.1
Cancel the common factor.
Step 8.2.2.1.2
Divide by .
Step 8.2.3
Simplify the right side.
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Step 8.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 8.2.3.2
Multiply .
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Step 8.2.3.2.1
Multiply by .
Step 8.2.3.2.2
Multiply by .
Step 9
Find the period of .
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Step 9.1
The period of the function can be calculated using .
Step 9.2
Replace with in the formula for period.
Step 9.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 11
Consolidate the answers.
, for any integer