Trigonometry Examples

Solve for x 2sin(x)+cot(x)-csc(x)=0
Step 1
Simplify the left side.
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Step 1.1
Simplify each term.
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Step 1.1.1
Rewrite in terms of sines and cosines.
Step 1.1.2
Rewrite in terms of sines and cosines.
Step 2
Multiply both sides of the equation by .
Step 3
Apply the distributive property.
Step 4
Simplify.
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Step 4.1
Rewrite using the commutative property of multiplication.
Step 4.2
Cancel the common factor of .
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Step 4.2.1
Cancel the common factor.
Step 4.2.2
Rewrite the expression.
Step 4.3
Rewrite using the commutative property of multiplication.
Step 5
Simplify each term.
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Step 5.1
Multiply .
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Step 5.1.1
Raise to the power of .
Step 5.1.2
Raise to the power of .
Step 5.1.3
Use the power rule to combine exponents.
Step 5.1.4
Add and .
Step 5.2
Cancel the common factor of .
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Step 5.2.1
Factor out of .
Step 5.2.2
Cancel the common factor.
Step 5.2.3
Rewrite the expression.
Step 6
Multiply by .
Step 7
Replace the with based on the identity.
Step 8
Simplify each term.
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Step 8.1
Apply the distributive property.
Step 8.2
Multiply by .
Step 8.3
Multiply by .
Step 9
Subtract from .
Step 10
Substitute for .
Step 11
Factor the left side of the equation.
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Step 11.1
Factor out of .
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Step 11.1.1
Factor out of .
Step 11.1.2
Factor out of .
Step 11.1.3
Rewrite as .
Step 11.1.4
Factor out of .
Step 11.1.5
Factor out of .
Step 11.2
Factor.
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Step 11.2.1
Factor by grouping.
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Step 11.2.1.1
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 11.2.1.1.1
Factor out of .
Step 11.2.1.1.2
Rewrite as plus
Step 11.2.1.1.3
Apply the distributive property.
Step 11.2.1.1.4
Multiply by .
Step 11.2.1.2
Factor out the greatest common factor from each group.
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Step 11.2.1.2.1
Group the first two terms and the last two terms.
Step 11.2.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 11.2.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 11.2.2
Remove unnecessary parentheses.
Step 12
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 13
Set equal to and solve for .
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Step 13.1
Set equal to .
Step 13.2
Solve for .
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Step 13.2.1
Subtract from both sides of the equation.
Step 13.2.2
Divide each term in by and simplify.
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Step 13.2.2.1
Divide each term in by .
Step 13.2.2.2
Simplify the left side.
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Step 13.2.2.2.1
Cancel the common factor of .
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Step 13.2.2.2.1.1
Cancel the common factor.
Step 13.2.2.2.1.2
Divide by .
Step 13.2.2.3
Simplify the right side.
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Step 13.2.2.3.1
Move the negative in front of the fraction.
Step 14
Set equal to and solve for .
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Step 14.1
Set equal to .
Step 14.2
Add to both sides of the equation.
Step 15
The final solution is all the values that make true.
Step 16
Substitute for .
Step 17
Set up each of the solutions to solve for .
Step 18
Solve for in .
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Step 18.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 18.2
Simplify the right side.
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Step 18.2.1
The exact value of is .
Step 18.3
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 18.4
Simplify .
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Step 18.4.1
To write as a fraction with a common denominator, multiply by .
Step 18.4.2
Combine fractions.
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Step 18.4.2.1
Combine and .
Step 18.4.2.2
Combine the numerators over the common denominator.
Step 18.4.3
Simplify the numerator.
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Step 18.4.3.1
Multiply by .
Step 18.4.3.2
Subtract from .
Step 18.5
Find the period of .
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Step 18.5.1
The period of the function can be calculated using .
Step 18.5.2
Replace with in the formula for period.
Step 18.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 18.5.4
Divide by .
Step 18.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 19
Solve for in .
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Step 19.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 19.2
Simplify the right side.
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Step 19.2.1
The exact value of is .
Step 19.3
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 19.4
Subtract from .
Step 19.5
Find the period of .
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Step 19.5.1
The period of the function can be calculated using .
Step 19.5.2
Replace with in the formula for period.
Step 19.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 19.5.4
Divide by .
Step 19.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 20
List all of the solutions.
, for any integer
Step 21
Consolidate the answers.
, for any integer