Trigonometry Examples

Solve for x 2sin(x)+cot(x)=csc(x)
Step 1
Simplify the left side.
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Step 1.1
Rewrite in terms of sines and cosines.
Step 2
Simplify the right side.
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Step 2.1
Rewrite in terms of sines and cosines.
Step 3
Multiply both sides of the equation by .
Step 4
Apply the distributive property.
Step 5
Rewrite using the commutative property of multiplication.
Step 6
Cancel the common factor of .
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Step 6.1
Cancel the common factor.
Step 6.2
Rewrite the expression.
Step 7
Multiply .
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Step 7.1
Raise to the power of .
Step 7.2
Raise to the power of .
Step 7.3
Use the power rule to combine exponents.
Step 7.4
Add and .
Step 8
Cancel the common factor of .
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Step 8.1
Cancel the common factor.
Step 8.2
Rewrite the expression.
Step 9
Subtract from both sides of the equation.
Step 10
Replace the with based on the identity.
Step 11
Simplify each term.
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Step 11.1
Apply the distributive property.
Step 11.2
Multiply by .
Step 11.3
Multiply by .
Step 12
Subtract from .
Step 13
Substitute for .
Step 14
Factor the left side of the equation.
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Step 14.1
Factor out of .
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Step 14.1.1
Factor out of .
Step 14.1.2
Factor out of .
Step 14.1.3
Rewrite as .
Step 14.1.4
Factor out of .
Step 14.1.5
Factor out of .
Step 14.2
Factor.
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Step 14.2.1
Factor by grouping.
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Step 14.2.1.1
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 14.2.1.1.1
Factor out of .
Step 14.2.1.1.2
Rewrite as plus
Step 14.2.1.1.3
Apply the distributive property.
Step 14.2.1.1.4
Multiply by .
Step 14.2.1.2
Factor out the greatest common factor from each group.
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Step 14.2.1.2.1
Group the first two terms and the last two terms.
Step 14.2.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 14.2.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 14.2.2
Remove unnecessary parentheses.
Step 15
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 16
Set equal to and solve for .
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Step 16.1
Set equal to .
Step 16.2
Solve for .
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Step 16.2.1
Subtract from both sides of the equation.
Step 16.2.2
Divide each term in by and simplify.
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Step 16.2.2.1
Divide each term in by .
Step 16.2.2.2
Simplify the left side.
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Step 16.2.2.2.1
Cancel the common factor of .
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Step 16.2.2.2.1.1
Cancel the common factor.
Step 16.2.2.2.1.2
Divide by .
Step 16.2.2.3
Simplify the right side.
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Step 16.2.2.3.1
Move the negative in front of the fraction.
Step 17
Set equal to and solve for .
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Step 17.1
Set equal to .
Step 17.2
Add to both sides of the equation.
Step 18
The final solution is all the values that make true.
Step 19
Substitute for .
Step 20
Set up each of the solutions to solve for .
Step 21
Solve for in .
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Step 21.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 21.2
Simplify the right side.
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Step 21.2.1
The exact value of is .
Step 21.3
The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 21.4
Simplify .
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Step 21.4.1
To write as a fraction with a common denominator, multiply by .
Step 21.4.2
Combine fractions.
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Step 21.4.2.1
Combine and .
Step 21.4.2.2
Combine the numerators over the common denominator.
Step 21.4.3
Simplify the numerator.
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Step 21.4.3.1
Multiply by .
Step 21.4.3.2
Subtract from .
Step 21.5
Find the period of .
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Step 21.5.1
The period of the function can be calculated using .
Step 21.5.2
Replace with in the formula for period.
Step 21.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 21.5.4
Divide by .
Step 21.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 22
Solve for in .
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Step 22.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 22.2
Simplify the right side.
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Step 22.2.1
The exact value of is .
Step 22.3
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 22.4
Subtract from .
Step 22.5
Find the period of .
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Step 22.5.1
The period of the function can be calculated using .
Step 22.5.2
Replace with in the formula for period.
Step 22.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 22.5.4
Divide by .
Step 22.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 23
List all of the solutions.
, for any integer
Step 24
Consolidate the answers.
, for any integer