Trigonometry Examples

Solve for x sin(x)-cos(x)>0
Step 1
Divide each term in the equation by .
Step 2
Convert from to .
Step 3
Cancel the common factor of .
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Step 3.1
Cancel the common factor.
Step 3.2
Divide by .
Step 4
Separate fractions.
Step 5
Convert from to .
Step 6
Divide by .
Step 7
Multiply by .
Step 8
Add to both sides of the inequality.
Step 9
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 10
Simplify the right side.
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Step 10.1
The exact value of is .
Step 11
The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Step 12
Simplify .
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Step 12.1
To write as a fraction with a common denominator, multiply by .
Step 12.2
Combine fractions.
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Step 12.2.1
Combine and .
Step 12.2.2
Combine the numerators over the common denominator.
Step 12.3
Simplify the numerator.
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Step 12.3.1
Move to the left of .
Step 12.3.2
Add and .
Step 13
Find the period of .
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Step 13.1
The period of the function can be calculated using .
Step 13.2
Replace with in the formula for period.
Step 13.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 13.4
Divide by .
Step 14
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 15
Consolidate the answers.
, for any integer
Step 16
Use each root to create test intervals.
Step 17
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 17.1
Test a value on the interval to see if it makes the inequality true.
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Step 17.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 17.1.2
Replace with in the original inequality.
Step 17.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 17.2
Compare the intervals to determine which ones satisfy the original inequality.
True
True
Step 18
The solution consists of all of the true intervals.
, for any integer
Step 19