Ema Cinca (Y12)

Exploration of Trigonometry

The goal of our IB Analysis and Approaches Maths curriculum is to analyse abstract maths theories to train students in the discipline of pure maths. This term, we just finished learning about trigonometric identities. This involved an exploration into proving the identities from basic geometric shapes and other basic functions.

What trigonometric identities are is essentially a series of formulas that relate the values of trigonometric functions to each other. Its significance stems from its applications as a powerful tool for simplifying and solving complex trigonometric expressions.

Trigonometric identities allow us to manipulate trigonometric functions algebraically, which is crucial for solving problems in fields such as physics, engineering, and navigation. Moreover, trig identities help to establish important relationships between angles, which are useful in understanding the geometry of triangles, circles, and other shapes. They also have applications in computer graphics and animation, where they are used to create realistic simulations of three-dimensional objects and their movements.

Trigonometric identities can be derived in various ways, including using geometry, algebra, and complex analysis. One of the most common methods for proving trigonometric identities is by using the properties of the unit circle (aka a circle with a radius of 1 unit). Using the unit circle, we can derive trigonometric identities by applying trigonometric functions to the angles formed by the unit circle. For example, the cosine of an angle is equal to the x-coordinate of the point on the unit circle that corresponds to that angle, while the sine of an angle is equal to the y-coordinate of the same point. By using the properties of the unit circle, we can derive many trigonometric identities, such as the Pythagorean identity (sin2 +cos2=1) (see below for the mathematically written formula), the double angle identities and the compound identities.

In class, we spent several lessons deriving compound and double angle identities.

This image shows an example of some of my work, where we had to derive the compound angle identity of sin(A+B) using a rectangle leaning against a wall.

The main approach to a question like this is to first fill out a diagram like I did on the right with angles and sides derived from the data given.

Then you can form a triangle combining the two angles so that you can solve for sine as though it was a regular triangle.

Maths in an IB class is intended to challenge you, to explore how you can use given formulas to produce conclusions. Whether we know it or not, the process of inductive reasoning almost always is the way we form ideas about things.