# Trigonometry Examples

Solve over the Interval sec((3theta)/2)=-2 , [0,2pi)
,
Take the inverse secant of both sides of the equation to extract from inside the secant.
The exact value of is .
Multiply both sides of the equation by .
Simplify both sides of the equation.
Simplify the left side.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Multiply .
Multiply and .
Multiply by .
Multiply by .
The secant function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from to find the solution in the third quadrant.
Simplify the expression to find the second solution.
Multiply both sides of the equation by .
Simplify both sides of the equation.
Simplify the left side.
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Simplify .
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 .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Multiply .
Multiply and .
Multiply by .
Multiply by .
Find the period.
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
is approximately which is positive so remove the absolute value
Multiply the numerator by the reciprocal of the denominator.
Multiply .
Combine and .
Multiply by .
Combine and .
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 .
Plug in for and simplify to see if the solution is contained in .
Plug in for .
Simplify.
Simplify each term.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply .
Multiply by .
Multiply by .
The interval contains .
Plug in for and simplify to see if the solution is contained in .
Plug in for .
Simplify.
Simplify each term.
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
Multiply .
Multiply by .
Multiply by .
The interval contains .
Plug in for and simplify to see if the solution is contained in .
Plug in for .
Simplify.
Multiply by .
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 .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .