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Trigonometry Examples
Step 1
Subtract from both sides of the equation.
Step 2
Step 2.1
Since is an even function, rewrite as .
Step 2.2
Apply the distributive property.
Step 2.3
Multiply by .
Step 2.4
Multiply by .
Step 3
Replace the with based on the identity.
Step 4
Step 4.1
Apply the distributive property.
Step 4.2
Multiply by .
Step 4.3
Multiply by .
Step 5
Subtract from .
Step 6
Reorder the polynomial.
Step 7
Substitute for .
Step 8
Step 8.1
Factor out of .
Step 8.1.1
Factor out of .
Step 8.1.2
Factor out of .
Step 8.1.3
Rewrite as .
Step 8.1.4
Factor out of .
Step 8.1.5
Factor out of .
Step 8.2
Factor.
Step 8.2.1
Factor by grouping.
Step 8.2.1.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 8.2.1.1.1
Factor out of .
Step 8.2.1.1.2
Rewrite as plus
Step 8.2.1.1.3
Apply the distributive property.
Step 8.2.1.2
Factor out the greatest common factor from each group.
Step 8.2.1.2.1
Group the first two terms and the last two terms.
Step 8.2.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 8.2.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 8.2.2
Remove unnecessary parentheses.
Step 9
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 10
Step 10.1
Set equal to .
Step 10.2
Solve for .
Step 10.2.1
Add to both sides of the equation.
Step 10.2.2
Divide each term in by and simplify.
Step 10.2.2.1
Divide each term in by .
Step 10.2.2.2
Simplify the left side.
Step 10.2.2.2.1
Cancel the common factor of .
Step 10.2.2.2.1.1
Cancel the common factor.
Step 10.2.2.2.1.2
Divide by .
Step 11
Step 11.1
Set equal to .
Step 11.2
Add to both sides of the equation.
Step 12
The final solution is all the values that make true.
Step 13
Substitute for .
Step 14
Set up each of the solutions to solve for .
Step 15
Step 15.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 15.2
Simplify the right side.
Step 15.2.1
The exact value of is .
Step 15.3
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 15.4
Simplify .
Step 15.4.1
To write as a fraction with a common denominator, multiply by .
Step 15.4.2
Combine fractions.
Step 15.4.2.1
Combine and .
Step 15.4.2.2
Combine the numerators over the common denominator.
Step 15.4.3
Simplify the numerator.
Step 15.4.3.1
Multiply by .
Step 15.4.3.2
Subtract from .
Step 15.5
Find the period of .
Step 15.5.1
The period of the function can be calculated using .
Step 15.5.2
Replace with in the formula for period.
Step 15.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 15.5.4
Divide by .
Step 15.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 16
Step 16.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 16.2
Simplify the right side.
Step 16.2.1
The exact value of is .
Step 16.3
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 16.4
Subtract from .
Step 16.5
Find the period of .
Step 16.5.1
The period of the function can be calculated using .
Step 16.5.2
Replace with in the formula for period.
Step 16.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 16.5.4
Divide by .
Step 16.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 17
List all of the solutions.
, for any integer
Step 18
Consolidate and to .
, for any integer