# Trigonometry Examples

Solve over the Interval cos(2theta)=( square root of 3)/2 , [0,2pi)
,
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
The exact value of is .
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Multiply .
Multiply and .
Multiply by .
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.
Simplify the expression to find the second solution.
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.
Multiply by .
Subtract from .
Divide each term by and simplify.
Divide each term in by .
Cancel the common factor of .
Cancel the common factor.
Divide by .
Multiply .
Multiply and .
Multiply by .
Find the period.
The period of the function can be calculated using .
Replace with in the formula for period.
Solve the equation.
The absolute value is the distance between a number and zero. The distance between and is .
Cancel the common factor of .
Cancel the common factor.
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 .
Plug in for and simplify to see if the solution is contained in .
Plug in for .
Simplify.
Multiply by .
The interval contains .
Plug in for and simplify to see if the solution is contained in .
Plug in for .
Simplify.
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.
Move to the left of .
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.
Move to the left of .