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Trigonometry Examples
Step 1
Step 1.1
Simplify .
Step 1.1.1
Rewrite in terms of sines and cosines.
Step 1.1.2
Convert from to .
Step 2
Step 2.1
Simplify .
Step 2.1.1
Simplify the denominator.
Step 2.1.1.1
Rewrite in terms of sines and cosines.
Step 2.1.1.2
Apply the product rule to .
Step 2.1.2
Multiply the numerator by the reciprocal of the denominator.
Step 2.1.3
Apply the distributive property.
Step 2.1.4
Multiply .
Step 2.1.4.1
Combine and .
Step 2.1.4.2
Multiply by by adding the exponents.
Step 2.1.4.2.1
Multiply by .
Step 2.1.4.2.1.1
Raise to the power of .
Step 2.1.4.2.1.2
Use the power rule to combine exponents.
Step 2.1.4.2.2
Add and .
Step 2.1.5
Multiply by .
Step 2.1.6
Simplify each term.
Step 2.1.6.1
Factor out of .
Step 2.1.6.2
Multiply by .
Step 2.1.6.3
Separate fractions.
Step 2.1.6.4
Convert from to .
Step 2.1.6.5
Divide by .
Step 2.1.6.6
Convert from to .
Step 3
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 4
Replace the with based on the identity.
Step 5
Reorder the polynomial.
Step 6
Step 6.1
Move .
Step 6.2
Apply pythagorean identity.
Step 6.3
Add and .
Step 7
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 8
Step 8.1
Subtract from both sides of the equation.
Step 8.2
Simplify each term.
Step 8.2.1
Rewrite in terms of sines and cosines.
Step 8.2.2
Rewrite in terms of sines and cosines.
Step 8.2.3
Apply the product rule to .
Step 8.2.4
Combine and .
Step 8.2.5
Move the negative in front of the fraction.
Step 8.3
To write as a fraction with a common denominator, multiply by .
Step 8.4
To write as a fraction with a common denominator, multiply by .
Step 8.5
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Step 8.5.1
Multiply by .
Step 8.5.2
Multiply by .
Step 8.5.3
Reorder the factors of .
Step 8.6
Combine the numerators over the common denominator.
Step 8.7
Simplify the numerator.
Step 8.7.1
Factor out of .
Step 8.7.1.1
Factor out of .
Step 8.7.1.2
Factor out of .
Step 8.7.1.3
Factor out of .
Step 8.7.2
Apply the distributive property.
Step 8.7.3
Cancel the common factor of .
Step 8.7.3.1
Factor out of .
Step 8.7.3.2
Cancel the common factor.
Step 8.7.3.3
Rewrite the expression.
Step 8.7.4
Multiply by .
Step 8.8
Factor out of .
Step 8.9
Separate fractions.
Step 8.10
Convert from to .
Step 8.11
Convert from to .
Step 8.12
Combine and .
Step 8.13
Separate fractions.
Step 8.14
Rewrite in terms of sines and cosines.
Step 8.15
Rewrite as a product.
Step 8.16
Simplify.
Step 8.16.1
Convert from to .
Step 8.16.2
Convert from to .
Step 8.17
Multiply .
Step 8.17.1
Combine and .
Step 8.17.2
Combine and .
Step 8.18
Reorder factors in .
Step 9
Set the numerator equal to zero.
Step 10
Step 10.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 10.2
Set equal to and solve for .
Step 10.2.1
Set equal to .
Step 10.2.2
Solve for .
Step 10.2.2.1
Take the inverse cotangent of both sides of the equation to extract from inside the cotangent.
Step 10.2.2.2
Simplify the right side.
Step 10.2.2.2.1
The exact value of is .
Step 10.2.2.3
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 10.2.2.4
Simplify .
Step 10.2.2.4.1
To write as a fraction with a common denominator, multiply by .
Step 10.2.2.4.2
Combine fractions.
Step 10.2.2.4.2.1
Combine and .
Step 10.2.2.4.2.2
Combine the numerators over the common denominator.
Step 10.2.2.4.3
Simplify the numerator.
Step 10.2.2.4.3.1
Move to the left of .
Step 10.2.2.4.3.2
Add and .
Step 10.2.2.5
Find the period of .
Step 10.2.2.5.1
The period of the function can be calculated using .
Step 10.2.2.5.2
Replace with in the formula for period.
Step 10.2.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.2.2.5.4
Divide by .
Step 10.2.2.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 10.3
Set equal to and solve for .
Step 10.3.1
Set equal to .
Step 10.3.2
The range of cosecant is and . Since does not fall in this range, there is no solution.
No solution
No solution
Step 10.4
Set equal to and solve for .
Step 10.4.1
Set equal to .
Step 10.4.2
Solve for .
Step 10.4.2.1
Replace with .
Step 10.4.2.2
Solve for .
Step 10.4.2.2.1
Subtract from .
Step 10.4.2.2.2
Factor by grouping.
Step 10.4.2.2.2.1
Reorder terms.
Step 10.4.2.2.2.2
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 10.4.2.2.2.2.1
Factor out of .
Step 10.4.2.2.2.2.2
Rewrite as plus
Step 10.4.2.2.2.2.3
Apply the distributive property.
Step 10.4.2.2.2.2.4
Multiply by .
Step 10.4.2.2.2.2.5
Multiply by .
Step 10.4.2.2.2.3
Factor out the greatest common factor from each group.
Step 10.4.2.2.2.3.1
Group the first two terms and the last two terms.
Step 10.4.2.2.2.3.2
Factor out the greatest common factor (GCF) from each group.
Step 10.4.2.2.2.4
Factor the polynomial by factoring out the greatest common factor, .
Step 10.4.2.2.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 10.4.2.2.4
Set equal to and solve for .
Step 10.4.2.2.4.1
Set equal to .
Step 10.4.2.2.4.2
Solve for .
Step 10.4.2.2.4.2.1
Subtract from both sides of the equation.
Step 10.4.2.2.4.2.2
Divide each term in by and simplify.
Step 10.4.2.2.4.2.2.1
Divide each term in by .
Step 10.4.2.2.4.2.2.2
Simplify the left side.
Step 10.4.2.2.4.2.2.2.1
Dividing two negative values results in a positive value.
Step 10.4.2.2.4.2.2.2.2
Divide by .
Step 10.4.2.2.4.2.2.3
Simplify the right side.
Step 10.4.2.2.4.2.2.3.1
Divide by .
Step 10.4.2.2.4.2.3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 10.4.2.2.4.2.4
Simplify the right side.
Step 10.4.2.2.4.2.4.1
The exact value of is .
Step 10.4.2.2.4.2.5
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 10.4.2.2.4.2.6
Subtract from .
Step 10.4.2.2.4.2.7
Find the period of .
Step 10.4.2.2.4.2.7.1
The period of the function can be calculated using .
Step 10.4.2.2.4.2.7.2
Replace with in the formula for period.
Step 10.4.2.2.4.2.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.4.2.2.4.2.7.4
Divide by .
Step 10.4.2.2.4.2.8
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
, for any integer
Step 10.4.2.2.5
The final solution is all the values that make true.
, for any integer
, for any integer
, for any integer
, for any integer
Step 10.5
The final solution is all the values that make true.
, for any integer
, for any integer
Step 11
Step 11.1
Consolidate and to .
, for any integer
Step 11.2
Consolidate and to .
, for any integer
, for any integer
Step 12
Exclude the solutions that do not make true.
, for any integer