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Trigonometry

Trigonometry is a specialized branch of Geometry in which we pay excruciatingly close attention to the properties of triangles, in particular right triangles. Referring to Fig. 4.2 again, we define the sine of the angle $\theta$ (abbreviated $\sin \theta$) to be the ratio of the "far side" a to the hypotenuse c and the cosine of $\theta$ (abbreviated $\cos \theta$) to be the ratio of the "near side" b to the hypoteneuse c:

$$\sin \theta \equiv {a \over c} \qquad \qquad \cos \theta \equiv {b \over c}$$ (4.12)

The other trigonometric functions can easily be defined in terms of the $\sin$ and $\cos$:

tangent:  		 ${\displaystyle \tan \theta \equiv {a \over b} = {\sin \theta \over \cos \theta} }$ 
cotangent:  		 ${\displaystyle \cot \theta \equiv {b \over a} = {\sin \theta \over \cos \theta} = {1 \over \tan \theta} }$ 
secant:  		 ${\displaystyle \sec \theta \equiv {c \over b} = {1 \over \cos \theta} }$ 
cosecant:  		 ${\displaystyle \csc \theta \equiv {c \over a} = {1 \over \sin \theta} }$ 

For the life of me, I can't imagine why anyone invented the cotangent, the secant and the cosecant -- as far as I can tell, they are totally superfluous baggage that just slows you down in any actual calculations. Forget them. [Ahhhh. I have always wanted to say that! Of course you are wise enough to take my advice with a grain of salt, especially is you want to appear clever to Mathematicians....]

The sine and cosine of $\theta$ are our trigonometric workhorses. In no time at all, I will be wanting to think of them as functions - i.e. when you see " $\cos \theta$" I will want you to say, "cosine of theta" and think of it as $\cos(\theta)$ the same way you think of y(x). Whether as simple ratios or as functions, they have several delightful properties, the most important of which is obvious from the Pythagorean Theorem:^4.6

$$\cos^2 \theta + \sin^2 \theta = 1$$ (4.13)

where the notation $\sin^2 \theta$ means the square of $\sin \theta$ -- i.e. $\sin^2 \theta \equiv (\sin \theta) \times (\sin \theta)$ -- and similarly for $\cos \theta$. This convention is adopted to avoid confusion, believe it or not. If we wrote " $\sin \theta \; ^2$" it would be impossible to know for sure whether we meant $\sin(\theta^2)$ or $(\sin \theta)^2$; we could always put parentheses in the right places to remove the ambiguity, but in this case there is a convention instead. (People always have conventions when they are tired of thinking!)

I will need other trigonometric identities later on, but they can wait - why introduce math until we need it? [I have made an obvious exception in this Chapter as a whole only to "jump start" your Mathematical language (re)training.]