Trigonometry Examples

Solve for x sin(x)cos(x)=cot(x)
Step 1
Simplify the right side.
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Step 1.1
Rewrite in terms of sines and cosines.
Step 2
Multiply both sides of the equation by .
Step 3
Multiply .
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Step 3.1
Raise to the power of .
Step 3.2
Raise to the power of .
Step 3.3
Use the power rule to combine exponents.
Step 3.4
Add and .
Step 4
Cancel the common factor of .
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Step 4.1
Cancel the common factor.
Step 4.2
Rewrite the expression.
Step 5
Subtract from both sides of the equation.
Step 6
Simplify .
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Step 6.1
Factor out of .
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Step 6.1.1
Factor out of .
Step 6.1.2
Factor out of .
Step 6.1.3
Factor out of .
Step 6.2
Reorder and .
Step 6.3
Rewrite as .
Step 6.4
Factor out of .
Step 6.5
Factor out of .
Step 6.6
Rewrite as .
Step 6.7
Apply pythagorean identity.
Step 6.8
Rewrite using the commutative property of multiplication.
Step 6.9
Multiply by by adding the exponents.
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Step 6.9.1
Move .
Step 6.9.2
Multiply by .
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Step 6.9.2.1
Raise to the power of .
Step 6.9.2.2
Use the power rule to combine exponents.
Step 6.9.3
Add and .
Step 7
Solve for .
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Step 7.1
Divide each term in by and simplify.
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Step 7.1.1
Divide each term in by .
Step 7.1.2
Simplify the left side.
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Step 7.1.2.1
Dividing two negative values results in a positive value.
Step 7.1.2.2
Divide by .
Step 7.1.3
Simplify the right side.
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Step 7.1.3.1
Divide by .
Step 7.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 7.3
Simplify .
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Step 7.3.1
Rewrite as .
Step 7.3.2
Pull terms out from under the radical, assuming real numbers.
Step 7.4
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 7.5
Simplify the right side.
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Step 7.5.1
The exact value of is .
Step 7.6
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 7.7
Simplify .
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Step 7.7.1
To write as a fraction with a common denominator, multiply by .
Step 7.7.2
Combine fractions.
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Step 7.7.2.1
Combine and .
Step 7.7.2.2
Combine the numerators over the common denominator.
Step 7.7.3
Simplify the numerator.
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Step 7.7.3.1
Multiply by .
Step 7.7.3.2
Subtract from .
Step 7.8
Find the period of .
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Step 7.8.1
The period of the function can be calculated using .
Step 7.8.2
Replace with in the formula for period.
Step 7.8.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 7.8.4
Divide by .
Step 7.9
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 8
Consolidate the answers.
, for any integer