Trigonometry Examples

Find All Complex Solutions sin(x)^2-cos(x)^2=0
Step 1
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3
Set equal to and solve for .
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Step 3.1
Set equal to .
Step 3.2
Solve for .
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Step 3.2.1
Divide each term in the equation by .
Step 3.2.2
Convert from to .
Step 3.2.3
Cancel the common factor of .
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Step 3.2.3.1
Cancel the common factor.
Step 3.2.3.2
Rewrite the expression.
Step 3.2.4
Separate fractions.
Step 3.2.5
Convert from to .
Step 3.2.6
Divide by .
Step 3.2.7
Multiply by .
Step 3.2.8
Subtract from both sides of the equation.
Step 3.2.9
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 3.2.10
Simplify the right side.
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Step 3.2.10.1
The exact value of is .
Step 3.2.11
The tangent function is negative in the second and fourth quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 3.2.12
Simplify the expression to find the second solution.
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Step 3.2.12.1
Add to .
Step 3.2.12.2
The resulting angle of is positive and coterminal with .
Step 3.2.13
Find the period of .
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Step 3.2.13.1
The period of the function can be calculated using .
Step 3.2.13.2
Replace with in the formula for period.
Step 3.2.13.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.2.13.4
Divide by .
Step 3.2.14
Add to every negative angle to get positive angles.
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Step 3.2.14.1
Add to to find the positive angle.
Step 3.2.14.2
To write as a fraction with a common denominator, multiply by .
Step 3.2.14.3
Combine fractions.
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Step 3.2.14.3.1
Combine and .
Step 3.2.14.3.2
Combine the numerators over the common denominator.
Step 3.2.14.4
Simplify the numerator.
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Step 3.2.14.4.1
Move to the left of .
Step 3.2.14.4.2
Subtract from .
Step 3.2.14.5
List the new angles.
Step 3.2.15
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 4
Set equal to and solve for .
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Step 4.1
Set equal to .
Step 4.2
Solve for .
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Step 4.2.1
Divide each term in the equation by .
Step 4.2.2
Convert from to .
Step 4.2.3
Cancel the common factor of .
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Step 4.2.3.1
Cancel the common factor.
Step 4.2.3.2
Divide by .
Step 4.2.4
Separate fractions.
Step 4.2.5
Convert from to .
Step 4.2.6
Divide by .
Step 4.2.7
Multiply by .
Step 4.2.8
Add to both sides of the equation.
Step 4.2.9
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 4.2.10
Simplify the right side.
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Step 4.2.10.1
The exact value of is .
Step 4.2.11
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 4.2.12
Simplify .
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Step 4.2.12.1
To write as a fraction with a common denominator, multiply by .
Step 4.2.12.2
Combine fractions.
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Step 4.2.12.2.1
Combine and .
Step 4.2.12.2.2
Combine the numerators over the common denominator.
Step 4.2.12.3
Simplify the numerator.
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Step 4.2.12.3.1
Move to the left of .
Step 4.2.12.3.2
Add and .
Step 4.2.13
Find the period of .
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Step 4.2.13.1
The period of the function can be calculated using .
Step 4.2.13.2
Replace with in the formula for period.
Step 4.2.13.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.2.13.4
Divide by .
Step 4.2.14
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 5
The final solution is all the values that make true.
, for any integer