This post shows how the core trigonometric definitions, relations and addition theorems can be simply and intuitively visualized.     In the above diagram, the 3 classic trigonometric identities are now obvious: $$ \sin^2 \theta + \cos^2 \theta = 1; \quad \quad 1+\tan^2 \theta = \sec^2 \theta; \quad  \quad 1+\cot^2 \theta = \csc^2 \theta $$ Note that most of us learnt that the definitions of $\sec, \csc, \cot$ were $1/\cos, 1/\sin, 1/\tan$, respectively. However, from a historical and geometrical point of view, this is not quite true.  Cosine is originally defined as the sine of the complementary angle, $90-\theta$. Tan is equal to the length of the tangent resulting from  a radial line at angle $\theta$, and $\sec$ is equal to the length of the secant derived from $\theta$. And from this, we directly have $\cot$ which is therefore the tangent of the complementary angle, and $\csc$ which is the secant of the complementary angle.    Here is a common alternative to the above version. In this version all those on the lower-right hand side of the radial line (sin, tan, sec) are associated with the angle $\theta$. And all those on the upper-left hand side of the radial line (cosine, cotan, cosec) are associated with the complementary angle $90-\theta$.   Note that in both of them you learn that the tan function is intimately related to the tangent of a circle, and not just the ratio “opposite over hypotenuse’.   Hyperbolic Trig Functions Unfortunately there is not a single diagram that as succinctly shows the analogous hyperbolic relationships  and identities   From this diagram, we can clearly see $ \cosh^2 \theta – \sinh^2 \theta = 1 $, but the other two hyperbolic trig identities are not so obvious.   Figure 3. Hyperbolic Functions. This presents a direct analogy of figure 2.   See here for some other good versions.   Addition Theorems Looking at the vertical sides, we have: $\sin(\alpha+\beta) = \sin \alpha \; \cos \beta + \cos \alpha \; \sin \beta$. And comparing the horizontal sides, we have: $\cos(\alpha+\beta) = \cos \alpha \; \cos \beta – \sin \alpha \; \sin \beta$.     The white triangle on the left-hand side shows: $$\tan (\alpha + \beta) = \frac{\tan \alpha +\tan \beta}{1-\tan \alpha \tan \beta} $$.     The following diagram assists in understanding why  $\cos 2 \alpha = \cos^2 \alpha – \sin^2 \alpha $$ and  $\sin^2 \alpha = 2 \sin \alpha \; \cos \alpha.$       My name is Martin Roberts. I have a PhD  in theoretical physics. I love maths and computing. I’m open to new opportunities – consulting, contract or full-time – so let’s have a chat on how we can work together! Come follow me on Twitter: @Techsparx! My other contact details can be found here. [mc4wp_form id=”1021″] Other Posts you may like A Simple Formula for Sequences and Series Lissajous Curves The Perimeter of an Ellipse Multiple Pendulums