I think you need to understand algebra 2 before you begin trigonometry
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No. College Algebra and Precalculus (or Trigonometry) are enough preparation to take Calculus.
You have to pass one math course in high school to get your diploma when you graduate, but if you pass these courses and also take trigonometry, you will receive an advanced regents diploma. But it's definitely better to pass both and also trigonometry.
No, you can't. Although similar in concepts, Pre-Calculus is more advanced than Algebra 2. Algebra 2 is taken between Algebra 1 and Geometry or after Geometry and before Pre-Calculus. The reason that you can't take both at the same time is because of the curriculum. Pre-Calculus does not spend nearly as much time on linear topics (linear equations, linear programming, etc.) as Algebra 2 does. Pre-Calculus also almost always is 2 courses in one: Pre-Calculus and Trigonometry. Algebra 2 has very little, if any, trig. Topics that they have in common are quadratics equations/functions, polynomial equations/functions, rational functions, exponential & logarithmic functions (sometimes these are not covered in Algebra 2), possibly conic sections in Algebra 2, definitely in Pre-Calculus, factoring, and probability/sequences/series/statistics. In addition to trigonometry, pre-calculus also covers polar and parametric topics (these will NEVER NEVER NEVER be seen in Algebra 2) and an introduction to limits. So, you must take Algebra 2 before pre-calculus. If you want to take 2 math courses in 1 year, try algebra 1 and geometry (not very common), algebra 2 and geometry (somewhat common), and some schools allow honors students with a solid A in Algebra 2 (assuming you took Algebra 2 before Geometry, this differs between schools) allow you to take geometry and pre-calculus in the same year. The study of proofs is not a major topic in pre-calculus, and proofs make up a majority of geometry.
In many schools, geometry comes before algebra 2. You can ask your school if you can take geo online or at your own time. If you plan to take algebra 2 and trig, it is helpful to take geo.
No. It is considered 9th grade math because algebra is usually taken in 9th grade