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Trigonometry Examples
Step 1
Step 1.1
Simplify each term.
Step 1.1.1
Rewrite in terms of sines and cosines.
Step 1.1.2
Simplify the denominator.
Step 1.1.2.1
Rewrite as .
Step 1.1.2.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.1.2.3
Simplify.
Step 1.1.2.3.1
Rewrite in terms of sines and cosines.
Step 1.1.2.3.2
Rewrite in terms of sines and cosines.
Step 1.1.3
Combine and .
Step 1.1.4
Multiply the numerator by the reciprocal of the denominator.
Step 1.1.5
Multiply by .
Step 1.1.6
Rewrite in terms of sines and cosines.
Step 2
Multiply both sides of the equation by .
Step 3
Apply the distributive property.
Step 4
Step 4.1
Factor out of .
Step 4.2
Cancel the common factor.
Step 4.3
Rewrite the expression.
Step 5
Rewrite using the commutative property of multiplication.
Step 6
Step 6.1
Combine and .
Step 6.2
Raise to the power of .
Step 6.3
Raise to the power of .
Step 6.4
Use the power rule to combine exponents.
Step 6.5
Add and .
Step 7
Multiply by .
Step 8
Multiply both sides of the equation by .
Step 9
Apply the distributive property.
Step 10
Step 10.1
Combine and .
Step 10.2
Raise to the power of .
Step 10.3
Raise to the power of .
Step 10.4
Use the power rule to combine exponents.
Step 10.5
Add and .
Step 11
Rewrite using the commutative property of multiplication.
Step 12
Step 12.1
Factor out of .
Step 12.2
Cancel the common factor.
Step 12.3
Rewrite the expression.
Step 13
Multiply by .
Step 14
Replace the with based on the identity.
Step 15
Step 15.1
Apply the distributive property.
Step 15.2
Multiply by .
Step 15.3
Multiply by .
Step 16
Subtract from .
Step 17
Reorder the polynomial.
Step 18
Subtract from both sides of the equation.
Step 19
Step 19.1
Divide each term in by .
Step 19.2
Simplify the left side.
Step 19.2.1
Cancel the common factor of .
Step 19.2.1.1
Cancel the common factor.
Step 19.2.1.2
Divide by .
Step 19.3
Simplify the right side.
Step 19.3.1
Dividing two negative values results in a positive value.
Step 20
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 21
Step 21.1
Rewrite as .
Step 21.2
Multiply by .
Step 21.3
Combine and simplify the denominator.
Step 21.3.1
Multiply by .
Step 21.3.2
Raise to the power of .
Step 21.3.3
Raise to the power of .
Step 21.3.4
Use the power rule to combine exponents.
Step 21.3.5
Add and .
Step 21.3.6
Rewrite as .
Step 21.3.6.1
Use to rewrite as .
Step 21.3.6.2
Apply the power rule and multiply exponents, .
Step 21.3.6.3
Combine and .
Step 21.3.6.4
Cancel the common factor of .
Step 21.3.6.4.1
Cancel the common factor.
Step 21.3.6.4.2
Rewrite the expression.
Step 21.3.6.5
Evaluate the exponent.
Step 21.4
Simplify the numerator.
Step 21.4.1
Combine using the product rule for radicals.
Step 21.4.2
Multiply by .
Step 22
Step 22.1
First, use the positive value of the to find the first solution.
Step 22.2
Next, use the negative value of the to find the second solution.
Step 22.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 23
Set up each of the solutions to solve for .
Step 24
Step 24.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 24.2
Simplify the right side.
Step 24.2.1
Evaluate .
Step 24.3
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 24.4
Solve for .
Step 24.4.1
Remove parentheses.
Step 24.4.2
Simplify .
Step 24.4.2.1
Multiply by .
Step 24.4.2.2
Subtract from .
Step 24.5
Find the period of .
Step 24.5.1
The period of the function can be calculated using .
Step 24.5.2
Replace with in the formula for period.
Step 24.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 24.5.4
Divide by .
Step 24.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 25
Step 25.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 25.2
Simplify the right side.
Step 25.2.1
Evaluate .
Step 25.3
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 25.4
Solve for .
Step 25.4.1
Remove parentheses.
Step 25.4.2
Simplify .
Step 25.4.2.1
Multiply by .
Step 25.4.2.2
Subtract from .
Step 25.5
Find the period of .
Step 25.5.1
The period of the function can be calculated using .
Step 25.5.2
Replace with in the formula for period.
Step 25.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 25.5.4
Divide by .
Step 25.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 26
List all of the solutions.
, for any integer
Step 27
Step 27.1
Consolidate and to .
, for any integer
Step 27.2
Consolidate and to .
, for any integer
, for any integer
Step 28
Exclude the solutions that do not make true.
No solution