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Trigonometry Examples
Step 1
Add to both sides of the equation.
Step 2
Step 2.1
Divide each term in by .
Step 2.2
Simplify the left side.
Step 2.2.1
Cancel the common factor of .
Step 2.2.1.1
Cancel the common factor.
Step 2.2.1.2
Divide by .
Step 2.3
Simplify the right side.
Step 2.3.1
Multiply by .
Step 2.3.2
Combine and simplify the denominator.
Step 2.3.2.1
Multiply by .
Step 2.3.2.2
Raise to the power of .
Step 2.3.2.3
Raise to the power of .
Step 2.3.2.4
Use the power rule to combine exponents.
Step 2.3.2.5
Add and .
Step 2.3.2.6
Rewrite as .
Step 2.3.2.6.1
Use to rewrite as .
Step 2.3.2.6.2
Apply the power rule and multiply exponents, .
Step 2.3.2.6.3
Combine and .
Step 2.3.2.6.4
Cancel the common factor of .
Step 2.3.2.6.4.1
Cancel the common factor.
Step 2.3.2.6.4.2
Rewrite the expression.
Step 2.3.2.6.5
Evaluate the exponent.
Step 3
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 4
Step 4.1
The exact value of is .
Step 5
Step 5.1
Divide each term in by .
Step 5.2
Simplify the left side.
Step 5.2.1
Cancel the common factor of .
Step 5.2.1.1
Cancel the common factor.
Step 5.2.1.2
Divide by .
Step 5.3
Simplify the right side.
Step 5.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 5.3.2
Multiply .
Step 5.3.2.1
Multiply by .
Step 5.3.2.2
Multiply by .
Step 6
The cotangent 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 7
Step 7.1
Simplify.
Step 7.1.1
To write as a fraction with a common denominator, multiply by .
Step 7.1.2
Combine and .
Step 7.1.3
Combine the numerators over the common denominator.
Step 7.1.4
Add and .
Step 7.1.4.1
Reorder and .
Step 7.1.4.2
Add and .
Step 7.2
Divide each term in by and simplify.
Step 7.2.1
Divide each term in by .
Step 7.2.2
Simplify the left side.
Step 7.2.2.1
Cancel the common factor of .
Step 7.2.2.1.1
Cancel the common factor.
Step 7.2.2.1.2
Divide by .
Step 7.2.3
Simplify the right side.
Step 7.2.3.1
Multiply the numerator by the reciprocal of the denominator.
Step 7.2.3.2
Multiply .
Step 7.2.3.2.1
Multiply by .
Step 7.2.3.2.2
Multiply by .
Step 8
Step 8.1
The period of the function can be calculated using .
Step 8.2
Replace with in the formula for period.
Step 8.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 9
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 10
Consolidate the answers.
, for any integer