Trigonometry Examples

Solve over the Interval square root of 3csc(theta)-2=0 , [0,2pi)
,
Add to both sides of the equation.
Divide each term by and simplify.
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Divide each term in by .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Simplify .
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Multiply by .
Combine and simplify the denominator.
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Multiply and .
Raise to the power of .
Raise to the power of .
Use the power rule to combine exponents.
Add and .
Rewrite as .
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Rewrite as .
Apply the power rule and multiply exponents, .
Combine and .
Cancel the common factor of .
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Cancel the common factor.
Divide by .
Evaluate the exponent.
Take the inverse cosecant of both sides of the equation to extract from inside the cosecant.
The exact value of is .
The cosecant function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from to find the solution in the second quadrant.
Simplify .
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To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
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Move to the left of .
Subtract from .
Find the period.
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The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
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The absolute value is the distance between a number and zero. The distance between and is .
Divide by .
The period of the function is so values will repeat every radians in both directions.
, for any integer
Find the values of that produce a value within the interval .
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Plug in for and simplify to see if the solution is contained in .
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Plug in for .
Simplify.
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Multiply .
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Multiply by .
Multiply by .
Add and .
The interval contains .
Plug in for and simplify to see if the solution is contained in .
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Plug in for .
Simplify.
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Multiply .
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Multiply by .
Multiply by .
Add and .
The interval contains .
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