Trigonometry Examples

Solve for x cos(x)csc(x)=2cos(x)
Step 1
Simplify the left side.
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Step 1.1
Simplify .
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Step 1.1.1
Rewrite in terms of sines and cosines.
Step 1.1.2
Combine and .
Step 2
Multiply both sides of the equation by .
Step 3
Cancel the common factor of .
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Step 3.1
Cancel the common factor.
Step 3.2
Rewrite the expression.
Step 4
Rewrite using the commutative property of multiplication.
Step 5
Apply the sine double-angle identity.
Step 6
Subtract from both sides of the equation.
Step 7
Simplify each term.
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Step 7.1
Apply the sine double-angle identity.
Step 7.2
Multiply by .
Step 8
Factor out of .
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Step 8.1
Factor out of .
Step 8.2
Factor out of .
Step 8.3
Factor out of .
Step 9
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 10
Set equal to and solve for .
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Step 10.1
Set equal to .
Step 10.2
Solve for .
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Step 10.2.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 10.2.2
Simplify the right side.
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Step 10.2.2.1
The exact value of is .
Step 10.2.3
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 10.2.4
Simplify .
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Step 10.2.4.1
To write as a fraction with a common denominator, multiply by .
Step 10.2.4.2
Combine fractions.
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Step 10.2.4.2.1
Combine and .
Step 10.2.4.2.2
Combine the numerators over the common denominator.
Step 10.2.4.3
Simplify the numerator.
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Step 10.2.4.3.1
Multiply by .
Step 10.2.4.3.2
Subtract from .
Step 10.2.5
Find the period of .
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Step 10.2.5.1
The period of the function can be calculated using .
Step 10.2.5.2
Replace with in the formula for period.
Step 10.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.2.5.4
Divide by .
Step 10.2.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 11
Set equal to and solve for .
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Step 11.1
Set equal to .
Step 11.2
Solve for .
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Step 11.2.1
Subtract from both sides of the equation.
Step 11.2.2
Divide each term in by and simplify.
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Step 11.2.2.1
Divide each term in by .
Step 11.2.2.2
Simplify the left side.
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Step 11.2.2.2.1
Cancel the common factor of .
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Step 11.2.2.2.1.1
Cancel the common factor.
Step 11.2.2.2.1.2
Divide by .
Step 11.2.2.3
Simplify the right side.
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Step 11.2.2.3.1
Dividing two negative values results in a positive value.
Step 11.2.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 11.2.4
Simplify the right side.
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Step 11.2.4.1
The exact value of is .
Step 11.2.5
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 11.2.6
Simplify .
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Step 11.2.6.1
To write as a fraction with a common denominator, multiply by .
Step 11.2.6.2
Combine fractions.
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Step 11.2.6.2.1
Combine and .
Step 11.2.6.2.2
Combine the numerators over the common denominator.
Step 11.2.6.3
Simplify the numerator.
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Step 11.2.6.3.1
Move to the left of .
Step 11.2.6.3.2
Subtract from .
Step 11.2.7
Find the period of .
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Step 11.2.7.1
The period of the function can be calculated using .
Step 11.2.7.2
Replace with in the formula for period.
Step 11.2.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 11.2.7.4
Divide by .
Step 11.2.8
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 12
The final solution is all the values that make true.
, for any integer
Step 13
Consolidate and to .
, for any integer