Trigonometry Examples

Solve for x 2tan(x)-sec(x)^2=0
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 3
Reorder the polynomial.
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 using the perfect square rule.
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Step 5.2.1
Rewrite as .
Step 5.2.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 5.2.3
Rewrite the polynomial.
Step 5.2.4
Factor using the perfect square trinomial rule , where and .
Step 6
Divide each term in by and simplify.
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Step 6.1
Divide each term in by .
Step 6.2
Simplify the left side.
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Step 6.2.1
Dividing two negative values results in a positive value.
Step 6.2.2
Divide by .
Step 6.3
Simplify the right side.
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Step 6.3.1
Divide by .
Step 7
Set the equal to .
Step 8
Add to both sides of the equation.
Step 9
Substitute for .
Step 10
Take the inverse tangent of both sides of the equation to extract from inside the tangent.
Step 11
Simplify the right side.
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Step 11.1
The exact value of is .
Step 12
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 13
Simplify .
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Step 13.1
To write as a fraction with a common denominator, multiply by .
Step 13.2
Combine fractions.
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Step 13.2.1
Combine and .
Step 13.2.2
Combine the numerators over the common denominator.
Step 13.3
Simplify the numerator.
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Step 13.3.1
Move to the left of .
Step 13.3.2
Add and .
Step 14
Find the period of .
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Step 14.1
The period of the function can be calculated using .
Step 14.2
Replace with in the formula for period.
Step 14.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 14.4
Divide by .
Step 15
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 16
Consolidate the answers.
, for any integer