Trigonometry Examples

Solve for ? (tan(x)-1)(sec(x)+1)=0
Step 1
Simplify the left side of the equation.
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Step 1.1
Expand using the FOIL Method.
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Step 1.1.1
Apply the distributive property.
Step 1.1.2
Apply the distributive property.
Step 1.1.3
Apply the distributive property.
Step 1.2
Simplify each term.
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Step 1.2.1
Multiply by .
Step 1.2.2
Rewrite as .
Step 1.2.3
Multiply by .
Step 2
Factor .
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Step 2.1
Factor out the greatest common factor from each group.
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Step 2.1.1
Group the first two terms and the last two terms.
Step 2.1.2
Factor out the greatest common factor (GCF) from each group.
Step 2.2
Factor the polynomial by factoring out the greatest common factor, .
Step 3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4
Set equal to and solve for .
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Step 4.1
Set equal to .
Step 4.2
Solve for .
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Step 4.2.1
Subtract from both sides of the equation.
Step 4.2.2
Take the inverse secant of both sides of the equation to extract from inside the secant.
Step 4.2.3
Simplify the right side.
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Step 4.2.3.1
The exact value of is .
Step 4.2.4
The secant 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 4.2.5
Subtract from .
Step 4.2.6
Find the period of .
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Step 4.2.6.1
The period of the function can be calculated using .
Step 4.2.6.2
Replace with in the formula for period.
Step 4.2.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 4.2.6.4
Divide by .
Step 4.2.7
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
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
Add to both sides of the equation.
Step 5.2.2
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 5.2.3
Simplify the right side.
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Step 5.2.3.1
The exact value of is .
Step 5.2.4
The tangent 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 5.2.5
Simplify .
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Step 5.2.5.1
To write as a fraction with a common denominator, multiply by .
Step 5.2.5.2
Combine fractions.
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Step 5.2.5.2.1
Combine and .
Step 5.2.5.2.2
Combine the numerators over the common denominator.
Step 5.2.5.3
Simplify the numerator.
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Step 5.2.5.3.1
Move to the left of .
Step 5.2.5.3.2
Add and .
Step 5.2.6
Find the period of .
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Step 5.2.6.1
The period of the function can be calculated using .
Step 5.2.6.2
Replace with in the formula for period.
Step 5.2.6.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 5.2.6.4
Divide by .
Step 5.2.7
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 6
The final solution is all the values that make true.
, for any integer
Step 7
Consolidate and to .
, for any integer