# Trigonometry Examples

Find the Intersection of the Inequalities sin(theta)>0 , cos(theta)<0
,
Simplify the first inequality.
Take the inverse sine of both sides of the equation to extract from inside the sine.
and
The exact value of is .
and
The sine 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.
and
Subtract from .
and
Find the period.
The period of the function can be calculated using .
and
Replace with in the formula for period.
and
Solve the equation.
The absolute value is the distance between a number and zero. The distance between and is .
and
Divide by .
and
and
and
The period of the function is so values will repeat every radians in both directions.
and
and
and
Simplify the second inequality.
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
and
The exact value of is .
and
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 .
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 .
Find the period.
The period of the function can be calculated using .
and
Replace with in the formula for period.
and
Solve the equation.
The absolute value is the distance between a number and zero. The distance between and is .
and
Divide by .
and
and
and
The period of the function is so values will repeat every radians in both directions.
, for any integer