Trigonometry Examples

Solve for x csc(x)*cos(x)=csc(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
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
Cancel the common factor of .
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Step 4.1
Cancel the common factor.
Step 4.2
Rewrite the expression.
Step 5
Cancel the common factor of .
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Step 5.1
Cancel the common factor.
Step 5.2
Rewrite the expression.
Step 6
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 7
Simplify the right side.
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Step 7.1
The exact value of is .
Step 8
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 9
Subtract from .
Step 10
Find the period of .
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Step 10.1
The period of the function can be calculated using .
Step 10.2
Replace with in the formula for period.
Step 10.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.4
Divide by .
Step 11
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 12
Consolidate the answers.
, for any integer