Trigonometry Examples

Solve for x 2cos(x)^2-cos(x)=1
Step 1
Substitute for .
Step 2
Subtract from both sides of the equation.
Step 3
Factor by grouping.
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Step 3.1
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 3.1.1
Factor out of .
Step 3.1.2
Rewrite as plus
Step 3.1.3
Apply the distributive property.
Step 3.1.4
Multiply by .
Step 3.2
Factor out the greatest common factor from each group.
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Step 3.2.1
Group the first two terms and the last two terms.
Step 3.2.2
Factor out the greatest common factor (GCF) from each group.
Step 3.3
Factor the polynomial by factoring out the greatest common factor, .
Step 4
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 5
Set equal to and solve for .
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Step 5.1
Set equal to .
Step 5.2
Solve for .
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Step 5.2.1
Subtract from both sides of the equation.
Step 5.2.2
Divide each term in by and simplify.
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Step 5.2.2.1
Divide each term in by .
Step 5.2.2.2
Simplify the left side.
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Step 5.2.2.2.1
Cancel the common factor of .
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Step 5.2.2.2.1.1
Cancel the common factor.
Step 5.2.2.2.1.2
Divide by .
Step 5.2.2.3
Simplify the right side.
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Step 5.2.2.3.1
Move the negative in front of the fraction.
Step 6
Set equal to and solve for .
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Step 6.1
Set equal to .
Step 6.2
Add to both sides of the equation.
Step 7
The final solution is all the values that make true.
Step 8
Substitute for .
Step 9
Set up each of the solutions to solve for .
Step 10
Solve for in .
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Step 10.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 10.2
Simplify the right side.
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Step 10.2.1
The exact value of is .
Step 10.3
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 10.4
Simplify .
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Step 10.4.1
To write as a fraction with a common denominator, multiply by .
Step 10.4.2
Combine fractions.
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Step 10.4.2.1
Combine and .
Step 10.4.2.2
Combine the numerators over the common denominator.
Step 10.4.3
Simplify the numerator.
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Step 10.4.3.1
Multiply by .
Step 10.4.3.2
Subtract from .
Step 10.5
Find the period of .
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Step 10.5.1
The period of the function can be calculated using .
Step 10.5.2
Replace with in the formula for period.
Step 10.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 10.5.4
Divide by .
Step 10.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 11
Solve for in .
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Step 11.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 11.2
Simplify the right side.
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Step 11.2.1
The exact value of is .
Step 11.3
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 11.4
Subtract from .
Step 11.5
Find the period of .
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Step 11.5.1
The period of the function can be calculated using .
Step 11.5.2
Replace with in the formula for period.
Step 11.5.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 11.5.4
Divide by .
Step 11.6
The period of the function is so values will repeat every radians in both directions.
, for any integer
, for any integer
Step 12
List all of the solutions.
, for any integer
Step 13
Consolidate the answers.
, for any integer