Precalculus 3: trigonometry
A total of 52 hours of lectures
Due to the size of precalculus, it is divided into four parts. This page describes the third of those four parts.
Prerequisites
- High-school mathematics, mainly arithmetic; solving simple equations (linear and quadratic) with one unknown; some (but very few) polynomial equations will be solved during the course, but you can skip them if you want to (no problem at all; it will not affect your understanding of trigonometry).
- Precalculus 1: Basic notions (mainly the concept of function and related concepts; sets; logic)
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You will get a very brief introduction to *complex numbers* in this course, just enough for presenting one of applications of trigonometry, and for a nice illustration of the formulas for sine and cosine of a sum
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Some basic geometry (angles, triangles, polygons); a crash course (6 hours) is provided in Section 2 of this course (can be skipped if you feel that you master basics of geometry)
Curriculum
Make sure that you check with your professor what parts of the course you will need for your exam. Such things vary from country to country, from university to university, and they can even vary from year to year at the same university.
Precalculus 3: trigonometry
Get the outline
A detailed list of all the lectures in part 3 of the course, including which theorems will be discussed and which problems will be solved. If you are looking for a particular kind of problem or a particular concept, this is where you should look first.
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Course Objectives & Outcomes
How to solve problems in trigonometry (illustrated with 215 solved problems), in both geometrical and functional contexts, and why these methods work.
Number pi: its definition as the ratio of the perimeter to the diameter of a disk, relation to the area of a disk, some geometrical approximations.
Exact values of trigonometric functions for angles of 15, 18, 30, 36, 45, 60, 72, 75, and 22.5 degrees: geometric derivations, and with help of formulas.
Degree vs radian: how to use proportions for recalculating degrees to radians and back; reference angles.
The definition of trigonometric (circular) functions (sin, cos, tan) for *any* real number using the unit circle in the coordinate system.
Periodic functions. Sinusoids: period, amplitude, phase shift, vertical shift. Transformations of graphs of trigonometric functions.
Various trigonometric identities with proofs, geometrical illustrations, and applications for problem solving.
Sum and Difference Identities for sine and cosine with proofs, geometrical illustrations, and applications.
Double (Half) Angles Identities with geometrical illustrations, proofs, and applications in problem solving.
Compositions of trigonometric functions with inverse trigonometric functions; identities involving inverse trigonometric functions.
De Moivre’s formula (positive natural powers of complex numbers) and its application to quick recreation of formulas for sine and cosine of multiples of angles.
You get a sneak peek into trigonometry in a future Calculus class (how some trigonometric formulas are used there).
You get a crash course in Euclidean geometry: angles, triangles, polygons, similar triangles (proportions), inscribed and circumscribed circles, bisectors, etc.
The geometric definitions (by ratios in right triangles) of three trigonometric functions (sin, cos, tan) and their reciprocals (secant, cosecant, cotangent).
Solving triangles (finding side lengths and measures of all angles, knowing some of them), both right and oblique, with help of trigonometry.
The functional definition of sine, cosine and tangent, with help of unit circle and circular movement; properties of these functions.
Reference Angles Theorem with proof (by geometrical illustration) and applications; supplementary identities and the complementary angle properties.
Pythagorean theorem and Pythagorean triples. Law of Cosines, Law of Sines: formulation, proofs, and applications in problem solving.
The Pythagorean Identities; Reciprocal Identities; Quotient Identities; Even/odd identities.
Sum To Product and Product To Sum Formulas for sine and cosine, with derivations and applications.
Inverse functions to sine, cosine and tangent, their definitions, properties and graphs.
Complex numbers and their trigonometric (polar) form; consequences of the Sum Identities for sin and cos for multiplication of complex numbers in polar form.
Trigonometric equations: various types and corresponding methods for solutions; depicting the solution sets on the graphs and on the unit circle.
You get a plethora of geometric illustrations, supporting your intuition and understanding of trigonometry.
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