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Trigonometry Examples
Step 1
Step 1.1
Set the argument in greater than to find where the expression is defined.
Step 1.2
Solve for .
Step 1.2.1
Cross multiply.
Step 1.2.1.1
Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.
Step 1.2.1.2
Simplify the left side.
Step 1.2.1.2.1
Multiply by .
Step 1.2.2
Rewrite so is on the left side of the inequality.
Step 1.2.3
To remove the radical on the left side of the inequality, square both sides of the inequality.
Step 1.2.4
Simplify each side of the inequality.
Step 1.2.4.1
Use to rewrite as .
Step 1.2.4.2
Simplify the left side.
Step 1.2.4.2.1
Simplify .
Step 1.2.4.2.1.1
Apply the product rule to .
Step 1.2.4.2.1.2
Raise to the power of .
Step 1.2.4.2.1.3
Multiply the exponents in .
Step 1.2.4.2.1.3.1
Apply the power rule and multiply exponents, .
Step 1.2.4.2.1.3.2
Cancel the common factor of .
Step 1.2.4.2.1.3.2.1
Cancel the common factor.
Step 1.2.4.2.1.3.2.2
Rewrite the expression.
Step 1.2.4.2.1.4
Simplify.
Step 1.2.4.2.1.5
Apply the distributive property.
Step 1.2.4.2.1.6
Multiply by .
Step 1.2.4.3
Simplify the right side.
Step 1.2.4.3.1
Raising to any positive power yields .
Step 1.2.5
Solve for .
Step 1.2.5.1
Add to both sides of the inequality.
Step 1.2.5.2
Divide each term in by and simplify.
Step 1.2.5.2.1
Divide each term in by .
Step 1.2.5.2.2
Simplify the left side.
Step 1.2.5.2.2.1
Cancel the common factor of .
Step 1.2.5.2.2.1.1
Cancel the common factor.
Step 1.2.5.2.2.1.2
Divide by .
Step 1.2.5.2.3
Simplify the right side.
Step 1.2.5.2.3.1
Divide by .
Step 1.2.6
Find the domain of .
Step 1.2.6.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 1.2.6.2
Add to both sides of the inequality.
Step 1.2.6.3
Set the denominator in equal to to find where the expression is undefined.
Step 1.2.6.4
Add to both sides of the equation.
Step 1.2.6.5
The domain is all values of that make the expression defined.
Step 1.2.7
The solution consists of all of the true intervals.
Step 1.3
Set the radicand in greater than or equal to to find where the expression is defined.
Step 1.4
Add to both sides of the inequality.
Step 1.5
Set the denominator in equal to to find where the expression is undefined.
Step 1.6
Add to both sides of the equation.
Step 1.7
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Interval Notation:
Set-Builder Notation:
Step 2
Step 2.1
Replace the variable with in the expression.
Step 2.2
Subtract from .
Step 2.3
The expression contains a division by . The expression is undefined.
Undefined
Step 3
The radical expression end point is .
Step 4
Step 4.1
Substitute the value into . In this case, the point is .
Step 4.1.1
Replace the variable with in the expression.
Step 4.1.2
Simplify the result.
Step 4.1.2.1
Simplify the numerator.
Step 4.1.2.1.1
Subtract from .
Step 4.1.2.1.2
Any root of is .
Step 4.1.2.2
Simplify the expression.
Step 4.1.2.2.1
Subtract from .
Step 4.1.2.2.2
Multiply by .
Step 4.1.2.3
Logarithm base of is .
Step 4.1.2.3.1
Rewrite as an equation.
Step 4.1.2.3.2
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and does not equal , then is equivalent to .
Step 4.1.2.3.3
Create equivalent expressions in the equation that all have equal bases.
Step 4.1.2.3.4
Since the bases are the same, the two expressions are only equal if the exponents are also equal.
Step 4.1.2.3.5
The variable is equal to .
Step 4.1.2.4
The final answer is .
Step 4.2
Substitute the value into . In this case, the point is .
Step 4.2.1
Replace the variable with in the expression.
Step 4.2.2
Simplify the result.
Step 4.2.2.1
Subtract from .
Step 4.2.2.2
Subtract from .
Step 4.2.2.3
The final answer is .
Step 4.3
The square root can be graphed using the points around the vertex
Step 5