Trigonometry Examples

Solve for x sin(x/2)=-1/2
Step 1
Take the inverse sine of both sides of the equation to extract from inside the sine.
Step 2
Simplify the right side.
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Step 2.1
The exact value of is .
Step 3
Multiply both sides of the equation by .
Step 4
Simplify both sides of the equation.
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Step 4.1
Simplify the left side.
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Step 4.1.1
Cancel the common factor of .
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Step 4.1.1.1
Cancel the common factor.
Step 4.1.1.2
Rewrite the expression.
Step 4.2
Simplify the right side.
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Step 4.2.1
Simplify .
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Step 4.2.1.1
Cancel the common factor of .
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Step 4.2.1.1.1
Move the leading negative in into the numerator.
Step 4.2.1.1.2
Factor out of .
Step 4.2.1.1.3
Cancel the common factor.
Step 4.2.1.1.4
Rewrite the expression.
Step 4.2.1.2
Move the negative in front of the fraction.
Step 5
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 6
Simplify the expression to find the second solution.
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Step 6.1
Subtract from .
Step 6.2
The resulting angle of is positive, less than , and coterminal with .
Step 6.3
Solve for .
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Step 6.3.1
Multiply both sides of the equation by .
Step 6.3.2
Simplify both sides of the equation.
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Step 6.3.2.1
Simplify the left side.
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Step 6.3.2.1.1
Cancel the common factor of .
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Step 6.3.2.1.1.1
Cancel the common factor.
Step 6.3.2.1.1.2
Rewrite the expression.
Step 6.3.2.2
Simplify the right side.
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Step 6.3.2.2.1
Cancel the common factor of .
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Step 6.3.2.2.1.1
Factor out of .
Step 6.3.2.2.1.2
Cancel the common factor.
Step 6.3.2.2.1.3
Rewrite the expression.
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
Add to every negative angle to get positive angles.
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Step 8.1
Add to to find the positive angle.
Step 8.2
To write as a fraction with a common denominator, multiply by .
Step 8.3
Combine fractions.
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Step 8.3.1
Combine and .
Step 8.3.2
Combine the numerators over the common denominator.
Step 8.4
Simplify the numerator.
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Step 8.4.1
Multiply by .
Step 8.4.2
Subtract from .
Step 8.5
List the new angles.
Step 9
The period of the function is so values will repeat every radians in both directions.
, for any integer