¶Right-Triangle Trigonometry

Trigonometry is the branch of mathematics that studies relationships between side lengths and angles of triangles. It emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies.

For any right-angled triangle, the primary trigonometric functions (Sine, Cosine, and Tangent) map a given angle $\theta$ to the ratio of two specific side lengths.

¶The Tangent Function

The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

¶Astronomical Parallax

Trigonometry is the foundational mathematical tool used in astrophysics to measure interstellar distances. Stellar parallax is the apparent shift in position of a nearby star against the background of distant objects when viewed from opposite ends of Earth's orbit.

By forming a right triangle where:

1. The Adjacent side is the unknown distance to the star ($d$).
2. The Opposite side is the baseline radius of Earth's orbit (1 Astronomical Unit, or AU).
3. $\theta$ is the measured parallax angle ($p$).

Astronomers can solve for the distance using the small-angle approximation of the tangent function: $d = \frac{1 \text{ AU}}{\tan(p)} \approx \frac{1 \text{ AU}}{p \text{ (in radians)}}$

This technique is responsible for defining the unit of the "parsec" (parallax second)—the distance at which a star exhibits a parallax of one arcsecond.

Reference: Carroll, B. W., & Ostlie, D. A. (2017). An Introduction to Modern Astrophysics (2nd ed.). Cambridge University Press.

¶References

• Trigonometry (I.M. Gelfand, Mark Saul)
• Euler's Introductio in analysin infinitorum (Leonhard Euler, 1748)
• MIT OCW: 18.01 Single Variable Calculus (Trig Review)