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Trigonometry Examples
Step 1
Step 1.1
To write as a fraction with a common denominator, multiply by .
Step 1.2
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 1.2.1
Multiply by .
Step 1.2.2
Reorder the factors of .
Step 1.3
Combine the numerators over the common denominator.
Step 1.4
Simplify the numerator.
Step 1.4.1
Multiply by .
Step 1.4.2
Add and .
Step 2
Multiply both sides by .
Step 3
Step 3.1
Simplify the left side.
Step 3.1.1
Simplify .
Step 3.1.1.1
Rewrite using the commutative property of multiplication.
Step 3.1.1.2
Cancel the common factor of .
Step 3.1.1.2.1
Factor out of .
Step 3.1.1.2.2
Cancel the common factor.
Step 3.1.1.2.3
Rewrite the expression.
Step 3.1.1.3
Cancel the common factor of .
Step 3.1.1.3.1
Cancel the common factor.
Step 3.1.1.3.2
Rewrite the expression.
Step 3.2
Simplify the right side.
Step 3.2.1
Simplify .
Step 3.2.1.1
Cancel the common factor of .
Step 3.2.1.1.1
Factor out of .
Step 3.2.1.1.2
Factor out of .
Step 3.2.1.1.3
Cancel the common factor.
Step 3.2.1.1.4
Rewrite the expression.
Step 3.2.1.2
Combine and .
Step 4
Step 4.1
Rewrite the equation as .
Step 4.2
Multiply both sides of the equation by .
Step 4.3
Simplify both sides of the equation.
Step 4.3.1
Simplify the left side.
Step 4.3.1.1
Simplify .
Step 4.3.1.1.1
Cancel the common factor of .
Step 4.3.1.1.1.1
Cancel the common factor.
Step 4.3.1.1.1.2
Rewrite the expression.
Step 4.3.1.1.2
Cancel the common factor of .
Step 4.3.1.1.2.1
Factor out of .
Step 4.3.1.1.2.2
Cancel the common factor.
Step 4.3.1.1.2.3
Rewrite the expression.
Step 4.3.2
Simplify the right side.
Step 4.3.2.1
Simplify .
Step 4.3.2.1.1
Cancel the common factor of .
Step 4.3.2.1.1.1
Factor out of .
Step 4.3.2.1.1.2
Cancel the common factor.
Step 4.3.2.1.1.3
Rewrite the expression.
Step 4.3.2.1.2
Multiply by .
Step 5
Step 5.1
Set the denominator in equal to to find where the expression is undefined.
Step 5.2
Divide each term in by and simplify.
Step 5.2.1
Divide each term in by .
Step 5.2.2
Simplify the left side.
Step 5.2.2.1
Cancel the common factor of .
Step 5.2.2.1.1
Cancel the common factor.
Step 5.2.2.1.2
Divide by .
Step 5.2.3
Simplify the right side.
Step 5.2.3.1
Divide by .
Step 5.3
The domain is all values of that make the expression defined.
Step 6
Use each root to create test intervals.
Step 7
Step 7.1
Test a value on the interval to see if it makes the inequality true.
Step 7.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 7.1.2
Replace with in the original inequality.
Step 7.1.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 7.2
Test a value on the interval to see if it makes the inequality true.
Step 7.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 7.2.2
Replace with in the original inequality.
Step 7.2.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 7.3
Test a value on the interval to see if it makes the inequality true.
Step 7.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 7.3.2
Replace with in the original inequality.
Step 7.3.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 7.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 8
The solution consists of all of the true intervals.
or
Step 9
The result can be shown in multiple forms.
Inequality Form:
Interval Notation:
Step 10