Question

find the general solution for sec^2 theta= 3
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Answer 1

Answer

A helpful technique for solving cos(3θ/2)=−1/2 is to think of 3θ/2 as some new variable, call it X=3θ/2. Now we're solving cos(X)=−1/2, but we have to pay attention to the original bounds,

0≤θ≤2π.

We need to relate them to our new variable, X=3θ/2. We can do that by multiplying all three "sides" by 3/2:

000≤θ≤2π≤32θ≤322π≤X≤3π.

So now you're just solving cos(X)=−1/2 for 0≤X≤3π. Thinking graphically,

we know that, like you said, X=2π/3. We can see though that also X=4π/3 will work, as well as going around one full revolution beyond 2π/3 (note that 4π/3 plus a full revolution of 2π won't work, since 4π/3 is more than π, and another revolution would take you past 3π).

Don't forget to convert your Xs back into θs, where θ=23X, since X=32θ.