Trigonometry Examples

Solve for X cos(X/2)-1=0
Step 1
Add to both sides of the equation.
Step 2
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 3
Simplify the right side.
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Step 3.1
The exact value of is .
Step 4
Set the numerator equal to zero.
Step 5
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 6
Solve for .
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Step 6.1
Multiply both sides of the equation by .
Step 6.2
Simplify both sides of the equation.
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Step 6.2.1
Simplify the left side.
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Step 6.2.1.1
Cancel the common factor of .
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Step 6.2.1.1.1
Cancel the common factor.
Step 6.2.1.1.2
Rewrite the expression.
Step 6.2.2
Simplify the right side.
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Step 6.2.2.1
Simplify .
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Step 6.2.2.1.1
Subtract from .
Step 6.2.2.1.2
Multiply by .
Step 7
Find the period of .
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Step 7.1
The period of the function can be calculated using .
Step 7.2
Replace with in the formula for period.
Step 7.3
is approximately which is positive so remove the absolute value
Step 7.4
Multiply the numerator by the reciprocal of the denominator.
Step 7.5
Multiply by .
Step 8
The period of the function is so values will repeat every radians in both directions.
, for any integer
Step 9
Consolidate the answers.
, for any integer