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Trigonometry Examples
Step 1
Step 1.1
Let . Substitute for all occurrences of .
Step 1.2
Factor by grouping.
Step 1.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 1.2.1.1
Factor out of .
Step 1.2.1.2
Rewrite as plus
Step 1.2.1.3
Apply the distributive property.
Step 1.2.2
Factor out the greatest common factor from each group.
Step 1.2.2.1
Group the first two terms and the last two terms.
Step 1.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 1.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 1.3
Replace all occurrences of with .
Step 2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3
Step 3.1
Set equal to .
Step 3.2
Solve for .
Step 3.2.1
Add to both sides of the equation.
Step 3.2.2
Divide each term in by and simplify.
Step 3.2.2.1
Divide each term in by .
Step 3.2.2.2
Simplify the left side.
Step 3.2.2.2.1
Cancel the common factor of .
Step 3.2.2.2.1.1
Cancel the common factor.
Step 3.2.2.2.1.2
Divide by .
Step 3.2.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 3.2.4
Simplify the right side.
Step 3.2.4.1
The exact value of is .
Step 3.2.5
Multiply both sides of the equation by .
Step 3.2.6
Simplify both sides of the equation.
Step 3.2.6.1
Simplify the left side.
Step 3.2.6.1.1
Cancel the common factor of .
Step 3.2.6.1.1.1
Cancel the common factor.
Step 3.2.6.1.1.2
Rewrite the expression.
Step 3.2.6.2
Simplify the right side.
Step 3.2.6.2.1
Cancel the common factor of .
Step 3.2.6.2.1.1
Factor out of .
Step 3.2.6.2.1.2
Cancel the common factor.
Step 3.2.6.2.1.3
Rewrite the expression.
Step 3.2.7
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 3.2.8
Solve for .
Step 3.2.8.1
Multiply both sides of the equation by .
Step 3.2.8.2
Simplify both sides of the equation.
Step 3.2.8.2.1
Simplify the left side.
Step 3.2.8.2.1.1
Cancel the common factor of .
Step 3.2.8.2.1.1.1
Cancel the common factor.
Step 3.2.8.2.1.1.2
Rewrite the expression.
Step 3.2.8.2.2
Simplify the right side.
Step 3.2.8.2.2.1
Simplify .
Step 3.2.8.2.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 3.2.8.2.2.1.2
Simplify terms.
Step 3.2.8.2.2.1.2.1
Combine and .
Step 3.2.8.2.2.1.2.2
Combine the numerators over the common denominator.
Step 3.2.8.2.2.1.2.3
Cancel the common factor of .
Step 3.2.8.2.2.1.2.3.1
Factor out of .
Step 3.2.8.2.2.1.2.3.2
Cancel the common factor.
Step 3.2.8.2.2.1.2.3.3
Rewrite the expression.
Step 3.2.8.2.2.1.3
Simplify the numerator.
Step 3.2.8.2.2.1.3.1
Move to the left of .
Step 3.2.8.2.2.1.3.2
Subtract from .
Step 3.2.9
Find the period of .
Step 3.2.9.1
The period of the function can be calculated using .
Step 3.2.9.2
Replace with in the formula for period.
Step 3.2.9.3
is approximately which is positive so remove the absolute value
Step 3.2.9.4
Multiply the numerator by the reciprocal of the denominator.
Step 3.2.9.5
Multiply by .
Step 3.2.10
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 4
Step 4.1
Set equal to .
Step 4.2
Solve for .
Step 4.2.1
Add to both sides of the equation.
Step 4.2.2
Divide each term in by and simplify.
Step 4.2.2.1
Divide each term in by .
Step 4.2.2.2
Simplify the left side.
Step 4.2.2.2.1
Cancel the common factor of .
Step 4.2.2.2.1.1
Cancel the common factor.
Step 4.2.2.2.1.2
Divide by .
Step 4.2.3
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 4.2.4
Simplify the right side.
Step 4.2.4.1
Evaluate .
Step 4.2.5
Multiply both sides of the equation by .
Step 4.2.6
Simplify both sides of the equation.
Step 4.2.6.1
Simplify the left side.
Step 4.2.6.1.1
Cancel the common factor of .
Step 4.2.6.1.1.1
Cancel the common factor.
Step 4.2.6.1.1.2
Rewrite the expression.
Step 4.2.6.2
Simplify the right side.
Step 4.2.6.2.1
Multiply by .
Step 4.2.7
The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Step 4.2.8
Solve for .
Step 4.2.8.1
Multiply both sides of the equation by .
Step 4.2.8.2
Simplify both sides of the equation.
Step 4.2.8.2.1
Simplify the left side.
Step 4.2.8.2.1.1
Cancel the common factor of .
Step 4.2.8.2.1.1.1
Cancel the common factor.
Step 4.2.8.2.1.1.2
Rewrite the expression.
Step 4.2.8.2.2
Simplify the right side.
Step 4.2.8.2.2.1
Simplify .
Step 4.2.8.2.2.1.1
Subtract from .
Step 4.2.8.2.2.1.2
Multiply by .
Step 4.2.9
Find the period of .
Step 4.2.9.1
The period of the function can be calculated using .
Step 4.2.9.2
Replace with in the formula for period.
Step 4.2.9.3
is approximately which is positive so remove the absolute value
Step 4.2.9.4
Multiply the numerator by the reciprocal of the denominator.
Step 4.2.9.5
Multiply by .
Step 4.2.10
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 5
The final solution is all the values that make true.
, for any integer