Trigonometry/Book 2 Introduction

In Book 1 we used algebraic formulas to work with triangles.

Vector AB from A to B

This book deepens the understanding of the many relationships between triangles and circles. It also shows how to tackle some harder trigonometric function identities. Apart from more algebra, the main additional mathematical tool we use and develop is vectors in 3D.

Starting with Vectors

A point $\displaystyle P$ in 3D space is given by three coordinates, $\displaystyle P=(x,y,z)$. This isn't that different from points in 2D, where the points are $\displaystyle (x,y)$.

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If we have two points, $\displaystyle P$ and $\displaystyle Q$, we may use a different notation to indicate their coordinates. $\displaystyle P_{x},P_{y},P_{z}$ for $\displaystyle P$, and $\displaystyle Q_{x},Q_{y},Q_{z}$ for $\displaystyle Q$.

When we are more interested in the relative displacement between two points, we tend to use $\displaystyle V$ and $\displaystyle W$ as our names, and we call $\displaystyle V$ and $\displaystyle W$ vectors.

What has this to do with trigonometry?

If we have vectors $\displaystyle V$ and $\displaystyle W$ trigonometry comes in when we calculate the following quantity, which is called 'the dot product' and written $\displaystyle V\cdot W$

$\displaystyle V\cdot W=V_{x}W_{x}+VyW_{y}+V_{z}W_{z}$

It turns out that this dot product is the product of the lengths of the two vectors multiplied by the cosine of the angle between the two vectors. We'll try this with two vectors that are at right angles to each other.

Example: Two Perpendicular Vectors We choose $\displaystyle V=(1,3,4)$ and $\displaystyle W=(6,-2,0)$ then $\displaystyle V\cdot W=V_{x}W_{x}+VyW_{y}+V_{z}W_{z}$ $\displaystyle =1\times 6+3\times -2+4\times 0=6-6+0=0$ Neither of the vectors has zero length, but the dot product is zero. That means that the cosine of the angle between the two vectors is zero, the angle between the two vectors is ninety degrees and so the two vectors are at right angles to each other.

Exercise: Dot product of Vector with itself The angle 'between a vector and itself' is zero. What is the cosine of that angle? The dot product of two vectors is "the product of the lengths of the two vectors multiplied by the cosine of the angle between the two vectors." If both vectors are the same vector, how can we simplify that statement? We also have: $\displaystyle V\cdot V=V_{x}V_{x}+VyV_{y}+V_{z}V_{z}=(V_{x})^{2}+(V_{y})^{2}+(V_{z})^{2}$ ..you can relate this to the Pythagorean theorem, in three dimensions of course. What we've just seen is that the formula for dot product is closely related to trigonometry.

Applications

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add some applications (and pictures) as for Book 1.

About Book 2

Book 2, like book 1, only deals with aspects of trigonometry that can be handled without calculus. However, the algebra moves at a brisker pace than in Book 1. The topics are less central to understanding Trigonometry than in book 1.