Trigonometry
Key Identities
| Property | Statement |
|---|---|
| Pythagorean | sin²θ+cos²θ=1 |
| Double angle | sin2θ=2sinθcosθ |
| Law of sines | a/sinA=b/sinB=c/sinC |
| Law of cosines | c²=a²+b²−2ab cosC |
Examples
Example 1. Find sin(30°).
Solution. 1/2.
In Depth
Trigonometry studies the relationships between angles and side lengths in triangles, then extends these to periodic functions defined on the entire real line. The six trig functions — sin, cos, tan, csc, sec, cot — are defined via the unit circle and satisfy a rich web of identities.
The angle addition formulas \(\sin(A+B)=\sin A\cos B+\cos A\sin B\) and \(\cos(A+B)=\cos A\cos B-\sin A\sin B\) are the source of all other identities. From them derive double-angle, half-angle, product-to-sum, and sum-to-product formulas.
Inverse trig functions (arcsin, arccos, arctan) recover angles from ratios. Their derivatives — \(d/dx[\arcsin x]=1/\sqrt{1-x^2}\), \(d/dx[\arctan x]=1/(1+x^2)\) — appear in integration formulas for expressions involving \(\sqrt{a^2-x^2}\) and \(a^2+x^2\).
Applications span navigation (bearing and distance), physics (wave motion, oscillations, AC circuits), signal processing (Fourier series decomposes any periodic signal into sines and cosines), and computer graphics (rotation matrices use trig functions).
Further Reading & Context
The study of trigonometry connects to many areas of mathematics and its applications. Understanding the foundational definitions and theorems provides the basis for advanced work in analysis, algebra, and applied mathematics.
Historical development: most mathematical concepts evolved over centuries, with contributions from mathematicians across many cultures. The modern axiomatic treatment provides rigor, while computational tools enable practical application.
In modern mathematics, this topic appears in graduate courses and research across pure and applied mathematics. Connections to computer science, physics, and engineering make it a versatile and important area of study. Mastery of the core results and techniques opens doors to research in number theory, analysis, geometry, and beyond.
Recommended next steps: work through the standard theorems with full proofs, explore the connections to related topics listed above, and practice with a variety of problems ranging from computational exercises to theoretical proofs. The interplay between different areas of mathematics is one of the subject's greatest rewards.