Trigonometry Examples

Solve over the Interval 2cos(x)^2-sin(x)-1=0 , [0,2pi)
,
Step 1
Replace the with based on the identity.
Step 2
Simplify each term.
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Step 2.1
Apply the distributive property.
Step 2.2
Multiply by .
Step 2.3
Multiply by .
Step 3
Subtract from .
Step 4
Substitute for .
Step 5
Factor the left side of the equation.
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Step 5.1
Factor out of .
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Step 5.1.1
Factor out of .
Step 5.1.2
Factor out of .
Step 5.1.3
Rewrite as .
Step 5.1.4
Factor out of .
Step 5.1.5
Factor out of .
Step 5.2
Factor.
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Step 5.2.1
Factor by grouping.
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Step 5.2.1.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 5.2.1.1.1
Multiply by .
Step 5.2.1.1.2
Rewrite as plus
Step 5.2.1.1.3
Apply the distributive property.
Step 5.2.1.2
Factor out the greatest common factor from each group.
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Step 5.2.1.2.1
Group the first two terms and the last two terms.
Step 5.2.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 5.2.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 5.2.2
Remove unnecessary parentheses.
Step 6
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 7
Set equal to and solve for .
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Step 7.1
Set equal to .
Step 7.2
Solve for .
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Step 7.2.1
Add to both sides of the equation.
Step 7.2.2
Divide each term in by and simplify.
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Step 7.2.2.1
Divide each term in by .
Step 7.2.2.2
Simplify the left side.
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Step 7.2.2.2.1
Cancel the common factor of .
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Step 7.2.2.2.1.1
Cancel the common factor.
Step 7.2.2.2.1.2
Divide by .
Step 8
Set equal to and solve for .
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Step 8.1
Set equal to .
Step 8.2
Subtract from both sides of the equation.
Step 9
The final solution is all the values that make true.
Step 10
Substitute for .
Step 11
Set up each of the solutions to solve for .
Step 12
Solve for in .
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Step 12.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 12.2
Simplify the right side.
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Step 12.2.1
The exact value of is .
Step 12.3
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 12.4
Simplify .
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Step 12.4.1
To write as a fraction with a common denominator, multiply by .
Step 12.4.2
Combine fractions.
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Step 12.4.2.1
Combine and .
Step 12.4.2.2
Combine the numerators over the common denominator.
Step 12.4.3
Simplify the numerator.
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Step 12.4.3.1
Move to the left of .
Step 12.4.3.2
Subtract from .
Step 12.5
Find the period of .
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Step 12.5.1
The period of the function can be calculated using .
Step 12.5.2
Replace with in the formula for period.
Step 12.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 12.5.4
Divide by .
Step 12.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 13
Solve for in .
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Step 13.1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 13.2
Simplify the right side.
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Step 13.2.1
The exact value of is .
Step 13.3
The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from , to find a reference angle. Next, add this reference angle to to find the solution in the third quadrant.
Step 13.4
Simplify the expression to find the second solution.
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Step 13.4.1
Subtract from .
Step 13.4.2
The resulting angle of is positive, less than , and coterminal with .
Step 13.5
Find the period of .
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Step 13.5.1
The period of the function can be calculated using .
Step 13.5.2
Replace with in the formula for period.
Step 13.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 13.5.4
Divide by .
Step 13.6
Add to every negative angle to get positive angles.
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Step 13.6.1
Add to to find the positive angle.
Step 13.6.2
To write as a fraction with a common denominator, multiply by .
Step 13.6.3
Combine fractions.
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Step 13.6.3.1
Combine and .
Step 13.6.3.2
Combine the numerators over the common denominator.
Step 13.6.4
Simplify the numerator.
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Step 13.6.4.1
Multiply by .
Step 13.6.4.2
Subtract from .
Step 13.6.5
List the new angles.
Step 13.7
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 14
List all of the solutions.
, for any integer
Step 15
Consolidate the answers.
, for any integer
Step 16
Find the values of that produce a value within the interval .
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Step 16.1
Plug in for and simplify to see if the solution is contained in .
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Step 16.1.1
Plug in for .
Step 16.1.2
Simplify.
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Step 16.1.2.1
Simplify each term.
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Step 16.1.2.1.1
Cancel the common factor of and .
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Step 16.1.2.1.1.1
Factor out of .
Step 16.1.2.1.1.2
Cancel the common factors.
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Step 16.1.2.1.1.2.1
Factor out of .
Step 16.1.2.1.1.2.2
Cancel the common factor.
Step 16.1.2.1.1.2.3
Rewrite the expression.
Step 16.1.2.1.1.2.4
Divide by .
Step 16.1.2.1.2
Multiply .
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Step 16.1.2.1.2.1
Multiply by .
Step 16.1.2.1.2.2
Multiply by .
Step 16.1.2.2
Add and .
Step 16.1.3
The interval contains .
Step 16.2
Plug in for and simplify to see if the solution is contained in .
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Step 16.2.1
Plug in for .
Step 16.2.2
Simplify.
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Step 16.2.2.1
Multiply by .
Step 16.2.2.2
To write as a fraction with a common denominator, multiply by .
Step 16.2.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 16.2.2.3.1
Multiply by .
Step 16.2.2.3.2
Multiply by .
Step 16.2.2.4
Combine the numerators over the common denominator.
Step 16.2.2.5
Simplify the numerator.
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Step 16.2.2.5.1
Multiply by .
Step 16.2.2.5.2
Add and .
Step 16.2.3
The interval contains .
Step 16.3
Plug in for and simplify to see if the solution is contained in .
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Step 16.3.1
Plug in for .
Step 16.3.2
Simplify.
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Step 16.3.2.1
Multiply by .
Step 16.3.2.2
To write as a fraction with a common denominator, multiply by .
Step 16.3.2.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 16.3.2.3.1
Multiply by .
Step 16.3.2.3.2
Multiply by .
Step 16.3.2.4
Combine the numerators over the common denominator.
Step 16.3.2.5
Simplify the numerator.
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Step 16.3.2.5.1
Multiply by .
Step 16.3.2.5.2
Add and .
Step 16.3.2.6
Cancel the common factor of and .
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Step 16.3.2.6.1
Factor out of .
Step 16.3.2.6.2
Cancel the common factors.
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Step 16.3.2.6.2.1
Factor out of .
Step 16.3.2.6.2.2
Cancel the common factor.
Step 16.3.2.6.2.3
Rewrite the expression.
Step 16.3.3
The interval contains .