For 0 ≤ ϴ < 2π, how many solutions are there to tan(StartFraction theta Over 2 EndFraction) = sin(ϴ)? Note: Do not include values that are undefined for tan or sin(ϴ).

3 solutions:

Step-by-step explanation:

So, first of all, we need to figure the angles that cannot be included in our answers out. The only function in the equation that isn't defined for some angles is so let's focus on that part of the equation first.

We know that:

therefore:

so we need to find the angles that will make the cos function equal to zero. So we get:

or

we can now start plugging values in for n:

if we plugged any value greater than 0, we would end up with an angle that is greater than so,  that's the only angle we cannot include in our answer set, so:

having said this, we can now start solving the equation:

we can start solving this equation by using the half angle formula, such a formula tells us the following:

so we can substitute it into our equation:

we can now multiply both sides of the equation by

so we get:

we can use the pythagorean identity to rewrite in terms of cos:

so we get:

we can subtract a 1 from both sides of the equation so we end up with:

and we can now add

to both sides of the equation so we get:

and we can solve this equation by factoring. We can factor to get:

and we can use the zero product property to solve this, so we get two equations:

Equation 1:

Equation 2:

we add a 1 to both sides of the equation so we get:

so we end up with three answers to this equation:

ab will be equal to

b. 153 students do not participate in sports and seventy-five 12th graders were surveyed