Trigonometry Examples

Solve for x (1-cos(x))/(sin(x)) = square root of ((1-cos(x))^2)/(1-cos(x)^2)
Step 1
Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 2
To remove the radical on the left side of the equation, square both sides of the equation.
Step 3
Simplify each side of the equation.
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Step 3.1
Use to rewrite as .
Step 3.2
Simplify the left side.
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Step 3.2.1
Simplify .
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Step 3.2.1.1
Multiply the exponents in .
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Step 3.2.1.1.1
Apply the power rule and multiply exponents, .
Step 3.2.1.1.2
Cancel the common factor of .
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Step 3.2.1.1.2.1
Cancel the common factor.
Step 3.2.1.1.2.2
Rewrite the expression.
Step 3.2.1.2
Simplify the denominator.
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Step 3.2.1.2.1
Rewrite as .
Step 3.2.1.2.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 3.2.1.3
Reduce the expression by cancelling the common factors.
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Step 3.2.1.3.1
Cancel the common factor of and .
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Step 3.2.1.3.1.1
Factor out of .
Step 3.2.1.3.1.2
Cancel the common factors.
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Step 3.2.1.3.1.2.1
Factor out of .
Step 3.2.1.3.1.2.2
Cancel the common factor.
Step 3.2.1.3.1.2.3
Rewrite the expression.
Step 3.2.1.3.2
Simplify.
Step 3.3
Simplify the right side.
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Step 3.3.1
Simplify .
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Step 3.3.1.1
Apply the product rule to .
Step 3.3.1.2
Multiply by .
Step 3.3.1.3
Separate fractions.
Step 3.3.1.4
Convert from to .
Step 3.3.1.5
Simplify the expression.
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Step 3.3.1.5.1
Divide by .
Step 3.3.1.5.2
Rewrite as .
Step 3.3.1.6
Expand using the FOIL Method.
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Step 3.3.1.6.1
Apply the distributive property.
Step 3.3.1.6.2
Apply the distributive property.
Step 3.3.1.6.3
Apply the distributive property.
Step 3.3.1.7
Simplify and combine like terms.
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Step 3.3.1.7.1
Simplify each term.
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Step 3.3.1.7.1.1
Multiply by .
Step 3.3.1.7.1.2
Multiply by .
Step 3.3.1.7.1.3
Multiply by .
Step 3.3.1.7.1.4
Multiply .
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Step 3.3.1.7.1.4.1
Multiply by .
Step 3.3.1.7.1.4.2
Multiply by .
Step 3.3.1.7.1.4.3
Raise to the power of .
Step 3.3.1.7.1.4.4
Raise to the power of .
Step 3.3.1.7.1.4.5
Use the power rule to combine exponents.
Step 3.3.1.7.1.4.6
Add and .
Step 3.3.1.7.2
Subtract from .
Step 3.3.1.8
Apply the distributive property.
Step 3.3.1.9
Simplify.
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Step 3.3.1.9.1
Multiply by .
Step 3.3.1.9.2
Rewrite in terms of sines and cosines.
Step 3.3.1.9.3
Apply the product rule to .
Step 3.3.1.9.4
One to any power is one.
Step 3.3.1.9.5
Combine and .
Step 3.3.1.10
Convert from to .
Step 4
Solve for .
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Step 4.1
Multiply both sides by .
Step 4.2
Simplify.
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Step 4.2.1
Simplify the left side.
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Step 4.2.1.1
Cancel the common factor of .
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Step 4.2.1.1.1
Cancel the common factor.
Step 4.2.1.1.2
Rewrite the expression.
Step 4.2.2
Simplify the right side.
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Step 4.2.2.1
Simplify .
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Step 4.2.2.1.1
Simplify each term.
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Step 4.2.2.1.1.1
Rewrite in terms of sines and cosines.
Step 4.2.2.1.1.2
Apply the product rule to .
Step 4.2.2.1.1.3
One to any power is one.
Step 4.2.2.1.1.4
Rewrite in terms of sines and cosines.
Step 4.2.2.1.1.5
Apply the product rule to .
Step 4.2.2.1.1.6
One to any power is one.
Step 4.2.2.1.1.7
Multiply .
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Step 4.2.2.1.1.7.1
Combine and .
Step 4.2.2.1.1.7.2
Combine and .
Step 4.2.2.1.1.8
Move the negative in front of the fraction.
Step 4.2.2.1.1.9
Rewrite in terms of sines and cosines.
Step 4.2.2.1.1.10
Apply the product rule to .
Step 4.2.2.1.2
Expand by multiplying each term in the first expression by each term in the second expression.
Step 4.2.2.1.3
Simplify terms.
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Step 4.2.2.1.3.1
Simplify each term.
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Step 4.2.2.1.3.1.1
Multiply by .
Step 4.2.2.1.3.1.2
Combine and .
Step 4.2.2.1.3.1.3
Multiply by .
Step 4.2.2.1.3.1.4
Multiply .
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Step 4.2.2.1.3.1.4.1
Combine and .
Step 4.2.2.1.3.1.4.2
Raise to the power of .
Step 4.2.2.1.3.1.4.3
Raise to the power of .
Step 4.2.2.1.3.1.4.4
Use the power rule to combine exponents.
Step 4.2.2.1.3.1.4.5
Add and .
Step 4.2.2.1.3.1.5
Multiply by .
Step 4.2.2.1.3.1.6
Multiply .
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Step 4.2.2.1.3.1.6.1
Combine and .
Step 4.2.2.1.3.1.6.2
Multiply by by adding the exponents.
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Step 4.2.2.1.3.1.6.2.1
Multiply by .
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Step 4.2.2.1.3.1.6.2.1.1
Raise to the power of .
Step 4.2.2.1.3.1.6.2.1.2
Use the power rule to combine exponents.
Step 4.2.2.1.3.1.6.2.2
Add and .
Step 4.2.2.1.3.2
Simplify terms.
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Step 4.2.2.1.3.2.1
Combine the numerators over the common denominator.
Step 4.2.2.1.3.2.2
Subtract from .
Step 4.2.2.1.3.2.3
Add and .
Step 4.2.2.1.4
Simplify the numerator.
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Step 4.2.2.1.4.1
Factor out the greatest common factor from each group.
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Step 4.2.2.1.4.1.1
Group the first two terms and the last two terms.
Step 4.2.2.1.4.1.2
Factor out the greatest common factor (GCF) from each group.
Step 4.2.2.1.4.2
Factor the polynomial by factoring out the greatest common factor, .
Step 4.2.2.1.4.3
Rewrite as .
Step 4.2.2.1.4.4
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 4.2.2.1.4.5
Combine exponents.
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Step 4.2.2.1.4.5.1
Raise to the power of .
Step 4.2.2.1.4.5.2
Raise to the power of .
Step 4.2.2.1.4.5.3
Use the power rule to combine exponents.
Step 4.2.2.1.4.5.4
Add and .
Step 4.3
Solve for .
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Step 4.3.1
Subtract from both sides of the equation.
Step 4.3.2
Simplify .
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Step 4.3.2.1
To write as a fraction with a common denominator, multiply by .
Step 4.3.2.2
Combine and .
Step 4.3.2.3
Combine the numerators over the common denominator.
Step 4.3.2.4
Simplify each term.
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Step 4.3.2.4.1
Simplify the numerator.
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Step 4.3.2.4.1.1
Rewrite as .
Step 4.3.2.4.1.2
Expand using the FOIL Method.
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Step 4.3.2.4.1.2.1
Apply the distributive property.
Step 4.3.2.4.1.2.2
Apply the distributive property.
Step 4.3.2.4.1.2.3
Apply the distributive property.
Step 4.3.2.4.1.3
Simplify and combine like terms.
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Step 4.3.2.4.1.3.1
Simplify each term.
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Step 4.3.2.4.1.3.1.1
Multiply by .
Step 4.3.2.4.1.3.1.2
Multiply by .
Step 4.3.2.4.1.3.1.3
Multiply by .
Step 4.3.2.4.1.3.1.4
Multiply .
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Step 4.3.2.4.1.3.1.4.1
Multiply by .
Step 4.3.2.4.1.3.1.4.2
Multiply by .
Step 4.3.2.4.1.3.1.4.3
Raise to the power of .
Step 4.3.2.4.1.3.1.4.4
Raise to the power of .
Step 4.3.2.4.1.3.1.4.5
Use the power rule to combine exponents.
Step 4.3.2.4.1.3.1.4.6
Add and .
Step 4.3.2.4.1.3.2
Subtract from .
Step 4.3.2.4.1.4
Apply the distributive property.
Step 4.3.2.4.1.5
Simplify.
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Step 4.3.2.4.1.5.1
Multiply by .
Step 4.3.2.4.1.5.2
Multiply by .
Step 4.3.2.4.1.6
Expand by multiplying each term in the first expression by each term in the second expression.
Step 4.3.2.4.1.7
Simplify each term.
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Step 4.3.2.4.1.7.1
Multiply by .
Step 4.3.2.4.1.7.2
Rewrite as .
Step 4.3.2.4.1.7.3
Multiply by .
Step 4.3.2.4.1.7.4
Multiply .
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Step 4.3.2.4.1.7.4.1
Raise to the power of .
Step 4.3.2.4.1.7.4.2
Raise to the power of .
Step 4.3.2.4.1.7.4.3
Use the power rule to combine exponents.
Step 4.3.2.4.1.7.4.4
Add and .
Step 4.3.2.4.1.7.5
Multiply by .
Step 4.3.2.4.1.7.6
Multiply by by adding the exponents.
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Step 4.3.2.4.1.7.6.1
Move .
Step 4.3.2.4.1.7.6.2
Multiply by .
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Step 4.3.2.4.1.7.6.2.1
Raise to the power of .
Step 4.3.2.4.1.7.6.2.2
Use the power rule to combine exponents.
Step 4.3.2.4.1.7.6.3
Add and .
Step 4.3.2.4.1.8
Add and .
Step 4.3.2.4.1.9
Subtract from .
Step 4.3.2.4.1.10
Move .
Step 4.3.2.4.1.11
Reorder and .
Step 4.3.2.4.1.12
Multiply by .
Step 4.3.2.4.1.13
Factor out of .
Step 4.3.2.4.1.14
Factor out of .
Step 4.3.2.4.1.15
Apply pythagorean identity.
Step 4.3.2.4.1.16
Multiply by by adding the exponents.
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Step 4.3.2.4.1.16.1
Multiply by .
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Step 4.3.2.4.1.16.1.1
Raise to the power of .
Step 4.3.2.4.1.16.1.2
Use the power rule to combine exponents.
Step 4.3.2.4.1.16.2
Add and .
Step 4.3.2.4.1.17
Combine the opposite terms in .
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Step 4.3.2.4.1.17.1
Subtract from .
Step 4.3.2.4.1.17.2
Subtract from .
Step 4.3.2.4.1.18
Rewrite as .
Step 4.3.2.4.1.19
Factor out of .
Step 4.3.2.4.1.20
Factor out of .
Step 4.3.2.4.1.21
Rewrite as .
Step 4.3.2.4.1.22
Apply pythagorean identity.
Step 4.3.2.4.2
Cancel the common factor of .
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Step 4.3.2.4.2.1
Cancel the common factor.
Step 4.3.2.4.2.2
Divide by .
Step 4.3.2.5
Subtract from .
Step 4.3.3
Since , the equation will always be true for any value of .
All real numbers
All real numbers
All real numbers
Step 5
The result can be shown in multiple forms.
All real numbers
Interval Notation: