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Trigonometry Examples
Step 1
Replace the with based on the identity.
Step 2
Step 2.1
Apply the distributive property.
Step 2.2
Multiply by .
Step 3
Add and .
Step 4
Step 4.1
Subtract from both sides of the equation.
Step 4.2
Subtract from .
Step 5
Subtract from both sides of the equation.
Step 6
Step 6.1
Divide each term in by .
Step 6.2
Simplify the left side.
Step 6.2.1
Cancel the common factor of .
Step 6.2.1.1
Cancel the common factor.
Step 6.2.1.2
Divide by .
Step 6.3
Simplify the right side.
Step 6.3.1
Divide by .
Step 7
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 8
Any root of is .
Step 9
Step 9.1
First, use the positive value of the to find the first solution.
Step 9.2
Next, use the negative value of the to find the second solution.
Step 9.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 10
Set up each of the solutions to solve for .
Step 11
Step 11.1
Take the inverse secant of both sides of the equation to extract from inside the secant.
Step 11.2
Simplify the right side.
Step 11.2.1
The exact value of is .
Step 11.3
The secant 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 11.4
Subtract from .
Step 11.5
Find the period of .
Step 11.5.1
The period of the function can be calculated using .
Step 11.5.2
Replace with in the formula for period.
Step 11.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 11.5.4
Divide by .
Step 11.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 12
Step 12.1
Take the inverse secant of both sides of the equation to extract from inside the secant.
Step 12.2
Simplify the right side.
Step 12.2.1
The exact value of is .
Step 12.3
The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Step 12.4
Subtract from .
Step 12.5
Find the period of .
Step 12.5.1
The period of the function can be calculated using .
Step 12.5.2
Replace with in the formula for period.
Step 12.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 12.5.4
Divide by .
Step 12.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 13
List all of the solutions.
, for any integer
Step 14
Consolidate the answers.
, for any integer