Trigonometry Examples

Solve for x -sin(4x)>0
Step 1
Divide each term in by and simplify.
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Step 1.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 1.2
Simplify the left side.
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Step 1.2.1
Dividing two negative values results in a positive value.
Step 1.2.2
Divide by .
Step 1.3
Simplify the right side.
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Step 1.3.1
Divide by .
Step 2
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 3
Simplify the right side.
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Step 3.1
The exact value of is .
Step 4
Divide each term in by and simplify.
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Step 4.1
Divide each term in by .
Step 4.2
Simplify the left side.
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Step 4.2.1
Cancel the common factor of .
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Step 4.2.1.1
Cancel the common factor.
Step 4.2.1.2
Divide by .
Step 4.3
Simplify the right side.
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Step 4.3.1
Divide by .
Step 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 6
Solve for .
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Step 6.1
Simplify.
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Step 6.1.1
Multiply by .
Step 6.1.2
Add and .
Step 6.2
Divide each term in by and simplify.
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Step 6.2.1
Divide each term in by .
Step 6.2.2
Simplify the left side.
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Step 6.2.2.1
Cancel the common factor of .
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Step 6.2.2.1.1
Cancel the common factor.
Step 6.2.2.1.2
Divide by .
Step 7
Find the period of .
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Step 7.1
The period of the function can be calculated using .
Step 7.2
Replace with in the formula for period.
Step 7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 7.4
Cancel the common factor of and .
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Step 7.4.1
Factor out of .
Step 7.4.2
Cancel the common factors.
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Step 7.4.2.1
Factor out of .
Step 7.4.2.2
Cancel the common factor.
Step 7.4.2.3
Rewrite the expression.
Step 8
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 9
Consolidate the answers.
, for any integer
Step 10
Use each root to create test intervals.
Step 11
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 11.1
Test a value on the interval to see if it makes the inequality true.
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Step 11.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 11.1.2
Replace with in the original inequality.
Step 11.1.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 11.2
Test a value on the interval to see if it makes the inequality true.
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Step 11.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 11.2.2
Replace with in the original inequality.
Step 11.2.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 11.3
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
True
Step 12
The solution consists of all of the true intervals.
, for any integer
Step 13