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Trigonometry Examples
Step 1
Step 1.1
Apply the tangent double-angle identity.
Step 1.2
Simplify the denominator.
Step 1.2.1
Rewrite as .
Step 1.2.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 2
Step 2.1
Factor out of .
Step 2.2
Factor out of .
Step 2.3
Factor out of .
Step 3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4
Step 4.1
Set equal to .
Step 4.2
Solve for .
Step 4.2.1
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 4.2.2
Simplify the right side.
Step 4.2.2.1
The exact value of is .
Step 4.2.3
The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from to find the solution in the fourth quadrant.
Step 4.2.4
Add and .
Step 4.2.5
Find the period of .
Step 4.2.5.1
The period of the function can be calculated using .
Step 4.2.5.2
Replace with in the formula for period.
Step 4.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.2.5.4
Divide by .
Step 4.2.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 5
Step 5.1
Set equal to .
Step 5.2
Solve for .
Step 5.2.1
Find the LCD of the terms in the equation.
Step 5.2.1.1
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
Step 5.2.1.2
The LCM of one and any expression is the expression.
Step 5.2.2
Multiply each term in by to eliminate the fractions.
Step 5.2.2.1
Multiply each term in by .
Step 5.2.2.2
Simplify the left side.
Step 5.2.2.2.1
Simplify each term.
Step 5.2.2.2.1.1
Cancel the common factor of .
Step 5.2.2.2.1.1.1
Cancel the common factor.
Step 5.2.2.2.1.1.2
Rewrite the expression.
Step 5.2.2.2.1.2
Expand using the FOIL Method.
Step 5.2.2.2.1.2.1
Apply the distributive property.
Step 5.2.2.2.1.2.2
Apply the distributive property.
Step 5.2.2.2.1.2.3
Apply the distributive property.
Step 5.2.2.2.1.3
Simplify and combine like terms.
Step 5.2.2.2.1.3.1
Simplify each term.
Step 5.2.2.2.1.3.1.1
Multiply by .
Step 5.2.2.2.1.3.1.2
Multiply by .
Step 5.2.2.2.1.3.1.3
Multiply by .
Step 5.2.2.2.1.3.1.4
Rewrite using the commutative property of multiplication.
Step 5.2.2.2.1.3.1.5
Multiply by by adding the exponents.
Step 5.2.2.2.1.3.1.5.1
Move .
Step 5.2.2.2.1.3.1.5.2
Multiply by .
Step 5.2.2.2.1.3.2
Add and .
Step 5.2.2.2.1.3.3
Add and .
Step 5.2.2.2.1.4
Apply the distributive property.
Step 5.2.2.2.1.5
Multiply by .
Step 5.2.2.2.1.6
Multiply .
Step 5.2.2.2.1.6.1
Multiply by .
Step 5.2.2.2.1.6.2
Multiply by .
Step 5.2.2.2.2
Subtract from .
Step 5.2.2.3
Simplify the right side.
Step 5.2.2.3.1
Expand using the FOIL Method.
Step 5.2.2.3.1.1
Apply the distributive property.
Step 5.2.2.3.1.2
Apply the distributive property.
Step 5.2.2.3.1.3
Apply the distributive property.
Step 5.2.2.3.2
Simplify and combine like terms.
Step 5.2.2.3.2.1
Simplify each term.
Step 5.2.2.3.2.1.1
Multiply by .
Step 5.2.2.3.2.1.2
Multiply by .
Step 5.2.2.3.2.1.3
Multiply by .
Step 5.2.2.3.2.1.4
Rewrite using the commutative property of multiplication.
Step 5.2.2.3.2.1.5
Multiply by by adding the exponents.
Step 5.2.2.3.2.1.5.1
Move .
Step 5.2.2.3.2.1.5.2
Multiply by .
Step 5.2.2.3.2.2
Add and .
Step 5.2.2.3.2.3
Add and .
Step 5.2.2.3.3
Multiply by .
Step 5.2.3
Solve the equation.
Step 5.2.3.1
Subtract from both sides of the equation.
Step 5.2.3.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 5.2.3.3
Rewrite as .
Step 5.2.3.4
The complete solution is the result of both the positive and negative portions of the solution.
Step 5.2.3.4.1
First, use the positive value of the to find the first solution.
Step 5.2.3.4.2
Next, use the negative value of the to find the second solution.
Step 5.2.3.4.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 5.2.4
Set up each of the solutions to solve for .
Step 5.2.5
Solve for in .
Step 5.2.5.1
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 5.2.5.2
The inverse tangent of is undefined.
Undefined
Undefined
Step 5.2.6
Solve for in .
Step 5.2.6.1
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 5.2.6.2
The inverse tangent of is undefined.
Undefined
Undefined
Step 5.2.7
List all of the solutions.
No solution
No solution
No solution
Step 6
The final solution is all the values that make true.
, for any integer
Step 7
Consolidate the answers.
, for any integer