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Trigonometry Examples
Step 1
Step 1.1
Divide each term in by .
Step 1.2
Simplify the left side.
Step 1.2.1
Cancel the common factor of .
Step 1.2.1.1
Cancel the common factor.
Step 1.2.1.2
Rewrite the expression.
Step 1.3
Simplify the right side.
Step 1.3.1
Separate fractions.
Step 1.3.2
Rewrite in terms of sines and cosines.
Step 1.3.3
Multiply by the reciprocal of the fraction to divide by .
Step 1.3.4
Write as a fraction with denominator .
Step 1.3.5
Cancel the common factor of .
Step 1.3.5.1
Cancel the common factor.
Step 1.3.5.2
Rewrite the expression.
Step 1.3.6
Divide by .
Step 2
Rewrite so is on the left side of the inequality.
Step 3
Step 3.1
Divide each term in by .
Step 3.2
Simplify the left side.
Step 3.2.1
Cancel the common factor of .
Step 3.2.1.1
Cancel the common factor.
Step 3.2.1.2
Divide by .
Step 4
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 5
Step 5.1
The exact value of is .
Step 6
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 7
Step 7.1
To write as a fraction with a common denominator, multiply by .
Step 7.2
Combine fractions.
Step 7.2.1
Combine and .
Step 7.2.2
Combine the numerators over the common denominator.
Step 7.3
Simplify the numerator.
Step 7.3.1
Multiply by .
Step 7.3.2
Subtract from .
Step 8
Step 8.1
The period of the function can be calculated using .
Step 8.2
Replace with in the formula for period.
Step 8.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 8.4
Divide by .
Step 9
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 10
Step 10.1
Set the argument in equal to to find where the expression is undefined.
, for any integer
Step 10.2
The domain is all values of that make the expression defined.
, for any integer
, for any integer
Step 11
Use each root to create test intervals.
Step 12
Step 12.1
Test a value on the interval to see if it makes the inequality true.
Step 12.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.1.2
Replace with in the original inequality.
Step 12.1.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 12.2
Test a value on the interval to see if it makes the inequality true.
Step 12.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.2.2
Replace with in the original inequality.
Step 12.2.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 12.3
Test a value on the interval to see if it makes the inequality true.
Step 12.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.3.2
Replace with in the original inequality.
Step 12.3.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 12.4
Test a value on the interval to see if it makes the inequality true.
Step 12.4.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.4.2
Replace with in the original inequality.
Step 12.4.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 12.5
Test a value on the interval to see if it makes the inequality true.
Step 12.5.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.5.2
Replace with in the original inequality.
Step 12.5.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 12.6
Test a value on the interval to see if it makes the inequality true.
Step 12.6.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.6.2
Replace with in the original inequality.
Step 12.6.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 12.7
Test a value on the interval to see if it makes the inequality true.
Step 12.7.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.7.2
Replace with in the original inequality.
Step 12.7.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 12.8
Compare the intervals to determine which ones satisfy the original inequality.
False
True
True
False
False
True
True
False
True
True
False
False
True
True
Step 13
The solution consists of all of the true intervals.
or or or , for any integer
Step 14
Combine the intervals.
, for any integer
Step 15