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Trigonometry Examples
Step 1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 2
Set equal to .
Step 3
Step 3.1
Set equal to .
Step 3.2
Solve for .
Step 3.2.1
Subtract from both sides of the equation.
Step 3.2.2
Divide each term in by and simplify.
Step 3.2.2.1
Divide each term in by .
Step 3.2.2.2
Simplify the left side.
Step 3.2.2.2.1
Dividing two negative values results in a positive value.
Step 3.2.2.2.2
Divide by .
Step 3.2.2.3
Simplify the right side.
Step 3.2.2.3.1
Divide by .
Step 3.2.3
To solve for , rewrite the equation using properties of logarithms.
Step 3.2.4
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 3.2.5
Rewrite the equation as .
Step 4
The final solution is all the values that make true.
Step 5
Step 5.1
Set the argument in greater than to find where the expression is defined.
Step 5.2
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
Determine if the inequality is true.
Step 7.1.3.1
The equation cannot be solved because it is undefined.
Step 7.1.3.2
The left side has no solution, which means that the given statement is false.
False
False
False
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 greater than the right side , which means that the given statement is always true.
True
True
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 not greater than the right side , which means that the given statement is false.
False
False
Step 7.4
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
False
True
False
Step 8
The solution consists of all of the true intervals.
Step 9
The result can be shown in multiple forms.
Inequality Form:
Interval Notation:
Step 10