IB AA: Geometry and Trigonometry

This chapter expands on basic trigonometry to solve a wider range of geometric problems. You will begin by applying trigonometry to circles, learning to calculate the length of an arc and the area of a sector using angles measured in radians. The focus then shifts to non-right-angled triangles, where you will use the Sine Rule and the Cosine Rule to find unknown sides and angles. You will also learn a new formula for the area of a triangle that uses the sine of an included angle.

The second part of the chapter introduces the unit circle as a way to define trigonometric functions for any angle, positive or negative. This leads to the study of the graphs of sine, cosine, and tangent functions, including their periodic nature, amplitude, and transformations. You will learn key trigonometric identities, which are equations that are true for all values of the variables.

By the end, you should be able to solve problems in both two and three dimensions by choosing the appropriate rule or formula. The most challenging parts are often solving trigonometric equations, which requires strong algebraic skills and a good understanding of the unit circle, and visualizing 3D problems. This chapter heavily relies on concepts from algebra and functions.

Formulas for IB AA: Geometry and Trigonometry

Length of a circular arc: $l = r\theta$ \theta is the central angle in radians, r is the radius.
Area of a circular sector: $A = \frac{1}{2}r^2\theta$ \theta is the central angle in radians, r is the radius.
Cosine Rule: $c^2 = a^2 + b^2 - 2ab \cos C$ Used to find a side (c) given two sides (a, b) and the included angle (C).
Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ Relates the sides of a triangle to the sines of their opposite angles.
Area of a triangle: $A = \frac{1}{2}ab \sin C$ C is the angle included between sides of length a and b.
Tangent Identity: $\tan \theta = \frac{\sin \theta}{\cos \theta}$ Defines tangent in terms of sine and cosine.
Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$ A fundamental identity derived from the unit circle.
Double Angle Formula for Sine: $\sin(2\theta) = 2\sin\theta\cos\theta$ Expresses the sine of a double angle in terms of a single angle.
Double Angle Formula for Cosine: $\cos(2\theta) = \cos^2\theta - \sin^2\theta$ Also written as 2\cos^2\theta - 1 or 1 - 2\sin^2\theta.