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Trigonometry Examples
Step 1
Step 1.1
Divide each term in by .
Step 1.2
Simplify the left side.
Step 1.2.1
Cancel the common factor of .
Step 1.2.1.1
Cancel the common factor.
Step 1.2.1.2
Rewrite the expression.
Step 1.2.2
Cancel the common factor of .
Step 1.2.2.1
Cancel the common factor.
Step 1.2.2.2
Rewrite the expression.
Step 1.3
Simplify the right side.
Step 1.3.1
Separate fractions.
Step 1.3.2
Rewrite in terms of sines and cosines.
Step 1.3.3
Rewrite as a product.
Step 1.3.4
Multiply by .
Step 1.3.5
Simplify the denominator.
Step 1.3.5.1
Raise to the power of .
Step 1.3.5.2
Raise to the power of .
Step 1.3.5.3
Use the power rule to combine exponents.
Step 1.3.5.4
Add and .
Step 1.3.6
Combine fractions.
Step 1.3.6.1
Combine.
Step 1.3.6.2
Multiply by .
Step 1.3.7
Multiply by .
Step 1.3.8
Separate fractions.
Step 1.3.9
Convert from to .
Step 1.3.10
Multiply by .
Step 1.3.11
Combine and .
Step 2
Rewrite the equation as .
Step 3
Multiply both sides of the equation by .
Step 4
Step 4.1
Simplify the left side.
Step 4.1.1
Cancel the common factor of .
Step 4.1.1.1
Cancel the common factor.
Step 4.1.1.2
Rewrite the expression.
Step 4.2
Simplify the right side.
Step 4.2.1
Multiply by .
Step 5
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 6
Step 6.1
First, use the positive value of the to find the first solution.
Step 6.2
Next, use the negative value of the to find the second solution.
Step 6.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 7
Set up each of the solutions to solve for .
Step 8
Step 8.1
Take the inverse secant of both sides of the equation to extract from inside the secant.
Step 8.2
Simplify the right side.
Step 8.2.1
The exact value of is .
Step 8.3
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 8.4
Simplify .
Step 8.4.1
To write as a fraction with a common denominator, multiply by .
Step 8.4.2
Combine fractions.
Step 8.4.2.1
Combine and .
Step 8.4.2.2
Combine the numerators over the common denominator.
Step 8.4.3
Simplify the numerator.
Step 8.4.3.1
Multiply by .
Step 8.4.3.2
Subtract from .
Step 8.5
Find the period of .
Step 8.5.1
The period of the function can be calculated using .
Step 8.5.2
Replace with in the formula for period.
Step 8.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 8.5.4
Divide by .
Step 8.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 9
Step 9.1
Take the inverse secant of both sides of the equation to extract from inside the secant.
Step 9.2
Simplify the right side.
Step 9.2.1
The exact value of is .
Step 9.3
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 9.4
Simplify .
Step 9.4.1
To write as a fraction with a common denominator, multiply by .
Step 9.4.2
Combine fractions.
Step 9.4.2.1
Combine and .
Step 9.4.2.2
Combine the numerators over the common denominator.
Step 9.4.3
Simplify the numerator.
Step 9.4.3.1
Multiply by .
Step 9.4.3.2
Subtract from .
Step 9.5
Find the period of .
Step 9.5.1
The period of the function can be calculated using .
Step 9.5.2
Replace with in the formula for period.
Step 9.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 9.5.4
Divide by .
Step 9.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 10
List all of the solutions.
, for any integer
Step 11
Consolidate the answers.
, for any integer