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:

$$\sin^2 \theta + \cos^2 \theta = 1; \quad 1+\tan^2 \theta = \sec^2 \theta; \quad 1+\cot^2 \theta = \csc^2 \theta$$.

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  and 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.

# Hyperbolic Trig Functions

This is a bit more complicated but still most hyperbolic identities are now clearer. (Note that the light gray line is the tangent line). From the above diagram the classic trigonometric identities are now obvious:

$$\cosh^2 \theta – \sinh^2 \theta = 1; \tanh^2 \theta + \sech^2 \theta =1$$.

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.