Trigonometry Examples

Solve for ? sin(x/2)=-1
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
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Step 4
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 5
Simplify the expression to find the second solution.
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Step 5.1
Subtract from .
Step 5.2
The resulting angle of is positive, less than , and coterminal with .
Step 5.3
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Step 6
Find the period of .
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Step 6.1
The period of the function can be calculated using .
Step 6.2
Replace with in the formula for period.
Step 6.3
is approximately which is positive so remove the absolute value
Step 6.4
Multiply the numerator by the reciprocal of the denominator.
Step 6.5
Multiply by .
Step 7
Add to every negative angle to get positive angles.
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Step 7.1
Add to to find the positive angle.
Step 7.2
Subtract from .
Step 7.3
List the new angles.
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