Trigonometry Examples

Find All Complex Solutions 1-sin(theta)=2cos(theta)^2
Step 1
Subtract from both sides of the equation.
Step 2
Replace the with based on the identity.
Step 3
Simplify each term.
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Step 3.1
Apply the distributive property.
Step 3.2
Multiply by .
Step 3.3
Multiply by .
Step 4
Subtract from .
Step 5
Reorder the polynomial.
Step 6
Substitute for .
Step 7
Factor by grouping.
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Step 7.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 7.1.1
Factor out of .
Step 7.1.2
Rewrite as plus
Step 7.1.3
Apply the distributive property.
Step 7.1.4
Multiply by .
Step 7.2
Factor out the greatest common factor from each group.
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Step 7.2.1
Group the first two terms and the last two terms.
Step 7.2.2
Factor out the greatest common factor (GCF) from each group.
Step 7.3
Factor the polynomial by factoring out the greatest common factor, .
Step 8
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 9
Set equal to and solve for .
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Step 9.1
Set equal to .
Step 9.2
Solve for .
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Step 9.2.1
Subtract from both sides of the equation.
Step 9.2.2
Divide each term in by and simplify.
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Step 9.2.2.1
Divide each term in by .
Step 9.2.2.2
Simplify the left side.
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Step 9.2.2.2.1
Cancel the common factor of .
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Step 9.2.2.2.1.1
Cancel the common factor.
Step 9.2.2.2.1.2
Divide by .
Step 9.2.2.3
Simplify the right side.
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Step 9.2.2.3.1
Move the negative in front of the fraction.
Step 10
Set equal to and solve for .
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Step 10.1
Set equal to .
Step 10.2
Add to both sides of the equation.
Step 11
The final solution is all the values that make true.
Step 12
Substitute for .
Step 13
Set up each of the solutions to solve for .
Step 14
Solve for in .
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Step 14.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 14.2
Simplify the right side.
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Step 14.2.1
The exact value of is .
Step 14.3
The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.
Step 14.4
Simplify the expression to find the second solution.
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Step 14.4.1
Subtract from .
Step 14.4.2
The resulting angle of is positive, less than , and coterminal with .
Step 14.5
Find the period of .
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Step 14.5.1
The period of the function can be calculated using .
Step 14.5.2
Replace with in the formula for period.
Step 14.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 14.5.4
Divide by .
Step 14.6
Add to every negative angle to get positive angles.
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Step 14.6.1
Add to to find the positive angle.
Step 14.6.2
To write as a fraction with a common denominator, multiply by .
Step 14.6.3
Combine fractions.
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Step 14.6.3.1
Combine and .
Step 14.6.3.2
Combine the numerators over the common denominator.
Step 14.6.4
Simplify the numerator.
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Step 14.6.4.1
Multiply by .
Step 14.6.4.2
Subtract from .
Step 14.6.5
List the new angles.
Step 14.7
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 15
Solve for in .
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Step 15.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 15.2
Simplify the right side.
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Step 15.2.1
The exact value of is .
Step 15.3
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 15.4
Simplify .
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Step 15.4.1
To write as a fraction with a common denominator, multiply by .
Step 15.4.2
Combine fractions.
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Step 15.4.2.1
Combine and .
Step 15.4.2.2
Combine the numerators over the common denominator.
Step 15.4.3
Simplify the numerator.
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Step 15.4.3.1
Move to the left of .
Step 15.4.3.2
Subtract from .
Step 15.5
Find the period of .
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Step 15.5.1
The period of the function can be calculated using .
Step 15.5.2
Replace with in the formula for period.
Step 15.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 15.5.4
Divide by .
Step 15.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 16
List all of the solutions.
, for any integer
Step 17
Consolidate the answers.
, for any integer