Trigonometry Examples

Solve for ? cos(-1/3)cos(x)>0
Step 1
Divide each term in by and simplify.
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Step 1.1
Divide each term in by .
Step 1.2
Simplify the left side.
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Step 1.2.1
Cancel the common factor of .
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Step 1.2.1.1
Cancel the common factor.
Step 1.2.1.2
Divide by .
Step 1.3
Simplify the right side.
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Step 1.3.1
Separate fractions.
Step 1.3.2
Convert from to .
Step 1.3.3
Divide by .
Step 1.3.4
Evaluate .
Step 1.3.5
Multiply by .
Step 2
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 3
Simplify the right side.
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Step 3.1
The exact value of is .
Step 4
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 5
Simplify .
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Step 5.1
To write as a fraction with a common denominator, multiply by .
Step 5.2
Combine fractions.
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Step 5.2.1
Combine and .
Step 5.2.2
Combine the numerators over the common denominator.
Step 5.3
Simplify the numerator.
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Step 5.3.1
Multiply by .
Step 5.3.2
Subtract from .
Step 6
Find the period of .
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Step 6.1
The period of the function can be calculated using .
Step 6.2
Replace with in the formula for period.
Step 6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 6.4
Divide by .
Step 7
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 8
Consolidate the answers.
, for any integer
Step 9
Use each root to create test intervals.
Step 10
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 10.1
Test a value on the interval to see if it makes the inequality true.
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Step 10.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 10.1.2
Replace with in the original inequality.
Step 10.1.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 10.2
Test a value on the interval to see if it makes the inequality true.
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Step 10.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 10.2.2
Replace with in the original inequality.
Step 10.2.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 10.3
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
True
Step 11
The solution consists of all of the true intervals.
, for any integer
Step 12