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Trigonometry Examples
Step 1
Step 1.1
Rewrite as .
Step 1.2
Let . Substitute for all occurrences of .
Step 1.3
Factor using the AC method.
Step 1.3.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 1.3.2
Write the factored form using these integers.
Step 1.4
Replace all occurrences of with .
Step 2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3
Step 3.1
Set equal to .
Step 3.2
Solve for .
Step 3.2.1
Add to both sides of the equation.
Step 3.2.2
Create equivalent expressions in the equation that all have equal bases.
Step 3.2.3
Since the bases are the same, then two expressions are only equal if the exponents are also equal.
Step 4
Step 4.1
Set equal to .
Step 4.2
Solve for .
Step 4.2.1
Add to both sides of the equation.
Step 4.2.2
Take the natural logarithm of both sides of the equation to remove the variable from the exponent.
Step 4.2.3
Expand by moving outside the logarithm.
Step 4.2.4
Simplify the right side.
Step 4.2.4.1
The natural logarithm of is .
Step 4.2.5
Divide each term in by and simplify.
Step 4.2.5.1
Divide each term in by .
Step 4.2.5.2
Simplify the left side.
Step 4.2.5.2.1
Cancel the common factor of .
Step 4.2.5.2.1.1
Cancel the common factor.
Step 4.2.5.2.1.2
Divide by .
Step 4.2.5.3
Simplify the right side.
Step 4.2.5.3.1
Divide by .
Step 5
The final solution is all the values that make true.