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Trigonometry Examples
Step 1
Step 1.1
Apply the product rule to .
Step 1.2
Raise to the power of .
Step 1.3
Raise to the power of .
Step 2
Step 2.1
Subtract from both sides of the equation.
Step 2.2
Write as a fraction with a common denominator.
Step 2.3
Combine the numerators over the common denominator.
Step 2.4
Subtract from .
Step 3
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 4
Step 4.1
Rewrite as .
Step 4.2
Simplify the numerator.
Step 4.2.1
Rewrite as .
Step 4.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 4.3
Simplify the denominator.
Step 4.3.1
Rewrite as .
Step 4.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 5
Step 5.1
First, use the positive value of the to find the first solution.
Step 5.2
Next, use the negative value of the to find the second solution.
Step 5.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 6
Set up each of the solutions to solve for .
Step 7
Step 7.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 7.2
Simplify the right side.
Step 7.2.1
Evaluate .
Step 7.3
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 7.4
Solve for .
Step 7.4.1
Remove parentheses.
Step 7.4.2
Simplify .
Step 7.4.2.1
Multiply by .
Step 7.4.2.2
Subtract from .
Step 7.5
Find the period of .
Step 7.5.1
The period of the function can be calculated using .
Step 7.5.2
Replace with in the formula for period.
Step 7.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 7.5.4
Divide by .
Step 7.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 8
Step 8.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 8.2
Simplify the right side.
Step 8.2.1
Evaluate .
Step 8.3
The cosine 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 8.4
Solve for .
Step 8.4.1
Remove parentheses.
Step 8.4.2
Simplify .
Step 8.4.2.1
Multiply by .
Step 8.4.2.2
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
List all of the solutions.
, for any integer
Step 10
Step 10.1
Consolidate and to .
, for any integer
Step 10.2
Consolidate and to .
, for any integer
, for any integer