Precalculus 3: Trigonometry

• How to solve problems in trigonometry (illustrated with 215 solved problems), in both geometrical and functional contexts, and why these methods work.
• You get a crash course in Euclidean geometry: angles, triangles, polygons, similar triangles (proportions), inscribed and circumscribed circles, bisectors, etc.
• 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.
• The geometric definitions (by ratios in right triangles) of three trigonometric functions (sin, cos, tan) and their reciprocals (secant, cosecant, cotangent).
• 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.
• Solving triangles (finding side lengths and measures of all angles, knowing some of them), both right and oblique, with help of trigonometry.
• Degree vs radian: how to use proportions for recalculating degrees to radians and back; reference angles.
• The functional definition of sine, cosine and tangent, with help of unit circle and circular movement; properties of these functions.
• The definition of trigonometric (circular) functions (sin, cos, tan) for *any* real number using the unit circle in the coordinate system.
• Reference Angles Theorem with proof (by geometrical illustration) and applications; supplementary identities and the complementary angle properties.
• Periodic functions. Sinusoids: period, amplitude, phase shift, vertical shift. Transformations of graphs of trigonometric functions.
• Pythagorean theorem and Pythagorean triples. Law of Cosines, Law of Sines: formulation, proofs, and applications in problem solving.
• Various trigonometric identities with proofs, geometrical illustrations, and applications for problem solving.
• The Pythagorean Identities; Reciprocal Identities; Quotient Identities; Even/odd identities.
• Sum and Difference Identities for sine and cosine with proofs, geometrical illustrations, and applications.
• Sum To Product and Product To Sum Formulas for sine and cosine, with derivations and applications.
• Double (Half) Angles Identities with geometrical illustrations, proofs, and applications in problem solving.
• Inverse functions to sine, cosine and tangent, their definitions, properties and graphs.
• Compositions of trigonometric functions with inverse trigonometric functions; identities involving inverse trigonometric functions.
• Complex numbers and their trigonometric (polar) form; consequences of the Sum Identities for sin and cos for multiplication of complex numbers in polar form.
• 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.
• Trigonometric equations: various types and corresponding methods for solutions; depicting the solution sets on the graphs and on the unit circle.
• You get a sneak peek into trigonometry in a future Calculus class (how some trigonometric formulas are used there).
• You get a plethora of geometric illustrations, supporting your intuition and understanding of trigonometry.

Precalculus 3: Trigonometry

Mathematics from high school to university

S1. Introduction to the course

You will learn: what is awaiting in this course, and what you are going to learn.

S2. Crash course in Euclidean geometry

You will learn: everything you need to know about geometry in order to feel comfortable with the new content in this course: geometrical concepts such as straight lines, straight line segments, angles, triangles (acute, right, obtuse), polygons, circles (inscribed, circumscribed), congruence rules for triangles (SSS, SAS, ASA), similar triangles, Thales' theorem, Pythagorean theorem, congruence rules for right triangles (HA, HL, LL), measuring angles, measuring distances, computing area of squares and triangles, isometries in the plane (symmetries, rotations, translations).

S3.14159... The magnificent number π

You will learn: about the number π: its meaning for circles and disks, and some basic (geometrical) approximation methods.

S4. Trigonometric functions of acute angles: the geometric approach

You will learn: the geometric definition of six trigonometric functions, why there are six of them, and how we can know that they are well defined as functions of (acute) angles; first (very basic) relationships between these functions.

S5. Computing exact values of trigonometric functions

You will learn: how to derive the exact values of trigonometric functions for angles: 15, 18, 30, 36, 45, 54, 60, 72, 75, and 22.5 degrees using geometric methods; we will also derive, also using just geometry, some trigonometric formulas valid for acute angles (but later, in the second half of the course, you will learn that all of them are valid just for any angle, so they are really worth learning); these formulas will be then used for computing values of trigonometric functions for some angles (knowing the values for some other angles). We will, step by step, create the graph of the sine and cosine functions for acute angles.

S6. An introduction to inverse trigonometric functions and to solving triangles

You will learn: the geometrical meaning of inverse trigonometric functions arcsine, arccosine and arctangent) for acute angles, and how to use them in simple problem solving (more advanced problem solving with triangles comes later in the course).

S7. From degrees to radians: why and how

You will learn: the definition of radian; how to calculate degrees to radians and back, using proportions; the values of the most common angles in radians; angles in the Cartesian coordinate system.

S8. Trigonometric (circular) functions of any angle: the unit circle and circular motion

You will learn: two ways of expanding the trigonometric functions sine and cosine (defined geometrically, for acute angles, in Section 3) to any angles (or, actually, to any real number):

[1] a static one: cos t = x, sin t = y, where (x,y) are the coordinates of the intersection point between the unit circle and the terminal side for the angle of t radians, in standard position (obviously functions R -> R as each point has exactly one pair of Cartesian coordinates),

[2] a dynamic one: a point is moving along the unit circle starting in the point (1,0) for t = 0, and continuing counterclockwise until the point on the circle where the length of the path from the beginning to this point is t; the coordinates of this point define the cosine and the sine functions as follows: x = cos t and y = sin t (obviously functions R -> R as each point has exactly one pair of Cartesian coordinates).

In order to construct these functions, we will wrap the number axis on the unit circle, which is a really cool operation.

S9. Basic properties of six trigonometric (circular) functions; graphing

You will learn: the definition of the other circular functions (tangent, and the three reciprocals) defined with help of sine and cosine; basic properties following immediately from the definitions and symmetries of the unit circle: the domain and range for all these functions, Reference Angles Identities, monotonicity in intervals, being even or odd, periodicity (a new concept, not introduced in Precalculus 1), the graphs; basic relationships between these functions: the Pythagorean Identity, cofunction identities. You will also learn the etymology of the names sine, tangent, and secant.

S10. Trigonometric identities; graph transformations

You will learn: good news for those who were afraid they were wasting their time in Section 5: everything done back there will be reused here! The only topic which must be redone is the derivation of the Sum Identities for sine and cosine, as the derivations done in Section 5 were geometrical and restricted to acute angles. All the other formulas (the double angle formulas, the power reduction formulas, half angle formulas, tangent half angle formulas, and triple angle formulas) were proven by formula manipulation, so they are valid also in the new situation. Two new groups of formulas (sum to product, and product to sum formulas). The Sum Identities will be used for graph transformations, which will also be discussed in this section. The terminology related to sinusoids will be introduced (period, phase, amplitude).

S11. Inverse trigonometric functions, their properties, and graphs

You will learn: about the inverse trigonometric functions arcsine, arccosine, and arctangent (the inverse to their reciprocals can be studied from the Precalculus book: pages 824-833; this is not covered in our course), their properties, graphs, and some interesting compositions with the trigonometric (circular) functions.

S12. More identities

You will learn: how to prove trigonometric identities.

S13. Trigonometric equations

You will learn: how to solve some basic types of trigonometric equations, how to write a series of solutions, and how to interpret both equations and their solution sets graphically. The following types of equations (or: methods of solving equations) are discussed:

[a] the very basic types of trigonometric equations: sin x = a, cos x = a, tan x = a,

[b] using sum or difference identities for sine and cosine,

[c] factorization: Sum-To-Product Formulas,

[d] factorization of polynomials,

[e] using the Product-To-Sum Formulas,

[f] reducing the degree of trigonometric functions,

[g] solution method by Universal Substitution: tangent of half argument,

[h] homogenous equations,

[i] combinations of the methods above.

S14. Some applications of trigonometry

You will learn: Including applications would make this course twice as large, so I will just concentrate on the most common applications. The lectures will not have the same level of detail as the lectures in the previous sections, but by now, you are probably able to read and understand Chapter 11 in the Precalculus book on your own, so do it and ask me questions if needed. I will address the following topics in this section:

[a] slopes of straight lines in the coordinate system,

[b] The Law of Cosines as a generalization of Pythagorean Theorem,

[c] a sine-based formula for the area of a triangle,

[d] The Law of Sines,

[e] Heron's Formula; solving oblique triangles,

[f] vectors in the plane (or in the 3-space) and angles between them,

[g] rotations and their matrices,

[h] complex numbers: rectangular and polar form,

[i] multiplication of complex numbers and an explanation of how its geometry is determined by the Sum Identities for the sine and cosine,

[j] de Moivre's formula for taking powers of complex numbers,

[k] roots of unity.

S15. Sneak peek into trigonometry in Calculus

You will learn: This section will give you some pointers to applications of trigonometry in Calculus. The purpose is not to teach you this stuff, but rather to give you an idea about how the skills gained during this course will help you in the future Calculus class. The topics mentioned here are:

[a] the limit of (sin x)/x in zero, and its importance in Calculus,

[b] the slope of a straight line and its importance for Differential Calculus,

[c] differentiability of the sine and cosine: which formulas to use,

[d] the derivatives (with examples of the sine, cosine, tangent, arcsine, and arctangent) and their role in finding extremums and for determining intervals of monotonicity,

[e] classes of functions (C^0, C^1, C^2, …) and some fun trigonometric examples,

[f] a word about Fourier and spirographs, Euler's formula, and Euler's identity,

[g] trigonometric functions in solutions of differential equations,

[h] polar coordinates in the plane,

[i] cylindrical and spherical coordinates,

[j] parametric curves,

[k] Power Reduction Formulas and integration,

[l] Trigonometric substitutions in integrals.

S16. Problem-solving: varia

You will learn: This section gives you a Smörgåsbord of problems to solve; the difficulty level varies, and, as the problems are not linked to specific sections, you will have to decide on your own what method to choose. Generally, the problems and exercises in the previous sections were on a basic level (with some minor exceptions), and the problems in this section are somewhat harder. Originally, I planned to assign badges Basic, Medium, or Hard to each problem, but then I thought: “Each problem you can't solve is hard; each problem you can solve is simple (for you).” So I changed my mind, and the problems are just presented to you without any labels.

S17. Extras

You will learn: about all the courses we offer. You will also get a glimpse into our plans for future courses, with approximate (very hypothetical!) release dates.

Make sure that you check with your professor what parts of the course you will need for your final 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.

A detailed description of the content of the course, with all the 208 videos and their titles, and with the texts of all the 215 problems solved during this course, is presented in the resource file

“001 List_of_all_Videos_and_Problems_Precalculus_3.pdf”

under video 1 ("Introduction to the course"). This content is also presented in video 1.