This post breaks down everything you need to understand the fundamentals of trigonometry, from the classic identities to vector-based triangle orientation. It’s a must-read for students starting precalculus, physics, or vector calculus.

Topics covered include:

• Core Trigonometric Identities:

  • sin(θ) = y/r

  • cos(θ) = x/r

  • tan(θ) = y/x = sin(θ)/cos(θ)

  • Their reciprocals: csc(θ), sec(θ), cot(θ)

  • Why these identities are clearer when interpreted geometrically using vectors from the origin.

• Vector Orientation in Trig:

  • A triangle positioned in the coordinate plane with side r as a vector

  • The relationship between sides x, y, and hypotenuse r:
    x² + y² = r²
    x = r·cos(θ), y = r·sin(θ)

• Inverse Trig Clarifications:

  • arctan(y/x) is not the reciprocal of tan(θ)

  • sin⁻¹(x) is an inverse function, not an exponent

• Natural Log and Function Composition:

  • ln(e^x) = x, and other useful inverse relationships

  • Understanding function inverses: f⁻¹(f(x)) = x

• Foundational Pythagorean Identity:

  • cos²(θ) + sin²(θ) = 1

  • All derived identities: sec²(θ) = 1 + tan²(θ), csc²(θ) = 1 + cot²(θ)

This is the core of trigonometry taught in most precalculus classes, simplified for intuitive understanding and visual learners.

This content is from the Ultimate Crash Course for STEM Majors, designed for students who want to build a rock-solid foundation in math with clean visuals and real reasoning—not just memorization.

Access the full 800+ page PDF crash course here:
https://author-jonathan-david-shop.fourthwall.com/products/the-ultimate-crash-course-includes-bonus-ultimate-cheat-sheet-800-pages-of-goodness

Browse all PDF books and bonus bundles:
https://author-jonathan-david-shop.fourthwall.com

Search tags: trigonometry identities chart, trig triangle vector graph, sin cos tan from unit circle, trig inverse vs reciprocal, difference between arctan and 1/tan, how to use trig with vectors, coordinate plane trig explanation, precalculus cheat sheet, trig review for exams.