Enter a problem...
Trigonometry Examples
Step 1
Step 1.1
Simplify each term.
Step 1.1.1
Rewrite in terms of sines and cosines, then cancel the common factors.
Step 1.1.1.1
Reorder and .
Step 1.1.1.2
Rewrite in terms of sines and cosines.
Step 1.1.1.3
Cancel the common factors.
Step 1.1.2
Rewrite in terms of sines and cosines.
Step 2
Step 2.1
Rewrite in terms of sines and cosines.
Step 3
Multiply both sides of the equation by .
Step 4
Apply the distributive property.
Step 5
Step 5.1
Raise to the power of .
Step 5.2
Raise to the power of .
Step 5.3
Use the power rule to combine exponents.
Step 5.4
Add and .
Step 6
Step 6.1
Cancel the common factor.
Step 6.2
Rewrite the expression.
Step 7
Step 7.1
Cancel the common factor.
Step 7.2
Rewrite the expression.
Step 8
Subtract from both sides of the equation.
Step 9
Step 9.1
Move .
Step 9.2
Reorder and .
Step 9.3
Rewrite as .
Step 9.4
Factor out of .
Step 9.5
Factor out of .
Step 9.6
Rewrite as .
Step 9.7
Apply pythagorean identity.
Step 10
Step 10.1
Factor the left side of the equation.
Step 10.1.1
Let . Substitute for all occurrences of .
Step 10.1.2
Factor out of .
Step 10.1.2.1
Factor out of .
Step 10.1.2.2
Raise to the power of .
Step 10.1.2.3
Factor out of .
Step 10.1.2.4
Factor out of .
Step 10.1.3
Replace all occurrences of with .
Step 10.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 10.3
Set equal to and solve for .
Step 10.3.1
Set equal to .
Step 10.3.2
Solve for .
Step 10.3.2.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 10.3.2.2
Simplify the right side.
Step 10.3.2.2.1
The exact value of is .
Step 10.3.2.3
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 10.3.2.4
Simplify .
Step 10.3.2.4.1
To write as a fraction with a common denominator, multiply by .
Step 10.3.2.4.2
Combine fractions.
Step 10.3.2.4.2.1
Combine and .
Step 10.3.2.4.2.2
Combine the numerators over the common denominator.
Step 10.3.2.4.3
Simplify the numerator.
Step 10.3.2.4.3.1
Multiply by .
Step 10.3.2.4.3.2
Subtract from .
Step 10.3.2.5
Find the period of .
Step 10.3.2.5.1
The period of the function can be calculated using .
Step 10.3.2.5.2
Replace with in the formula for period.
Step 10.3.2.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.3.2.5.4
Divide by .
Step 10.3.2.6
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 10.4
Set equal to and solve for .
Step 10.4.1
Set equal to .
Step 10.4.2
Solve for .
Step 10.4.2.1
Subtract from both sides of the equation.
Step 10.4.2.2
Divide each term in by and simplify.
Step 10.4.2.2.1
Divide each term in by .
Step 10.4.2.2.2
Simplify the left side.
Step 10.4.2.2.2.1
Dividing two negative values results in a positive value.
Step 10.4.2.2.2.2
Divide by .
Step 10.4.2.2.3
Simplify the right side.
Step 10.4.2.2.3.1
Divide by .
Step 10.4.2.3
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 10.4.2.4
Simplify the right side.
Step 10.4.2.4.1
The exact value of is .
Step 10.4.2.5
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 10.4.2.6
Subtract from .
Step 10.4.2.7
Find the period of .
Step 10.4.2.7.1
The period of the function can be calculated using .
Step 10.4.2.7.2
Replace with in the formula for period.
Step 10.4.2.7.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.4.2.7.4
Divide by .
Step 10.4.2.8
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 10.5
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