I personally would prefer a textbook recommendation I can download or pick up that is [preferably] not old and does not make trigonometry intimidating to approach (especially one that emphasizes understanding proofs behind properties/theorems).
I don't have textbooks to recommend, but I can recommend an approach to doing trigonometry that facilitates mathematical understanding of it by crystallizing the logical foundation of trigonometry and algebraic structure of trigonometric expressions. There are two 'levels' to this, depending on whether you want to go straight to complex numbers or stay within real trigonometry. In either case, the focus is on identifying the intrinsic core of trigonometry and reducing everything else to that.
Real trigonometry
The key quantities are $\cos(t)$ and $\sin(t)$, which are the $x$ and $y$ coordinates of the point $P_t$ on the unit circle that subtends an arc of length $t$ anti-clockwise from the $x$-axis, as depicted in the image from wikipedia:
Here arc length is measured along the unit circle, and $π$ is defined as the arc length of the semicircle, so $2π$ is $360°$. (This way of measuring angles is often called measuring them in "radians", but I personally think it is an unnecessary term.) Note that $P_t = P_{t+2πk}$ for any integer $k$, because $2πk$ would be an integer multiple of full rounds. Also note that increasing $t$ moves $P_t$ anti-clockwise, while decreasing $t$ moves $P_t$ clockwise. Related to that, $P_{-t}$ is the reflection of $P_t$ across the $x$-axis.
Note that the signs of $\cos(t)$ and $\sin(t)$ match exactly the signs of the $x$ and $y$ coordinates of the point on the circle. (Do not listen to people who tell you to memorize something to determine which of them is positive in which quadrant.)
And just by definition, $\cos(t)^2+\sin(t)^2 = 1$ for every real $t$. This is the first key algebraic fact.
Next, $\tan(t)$ is defined as $\sin(t)/\cos(t)$. (Historically, we also have defined $\sec(t) := 1/\cos(t)$ and $\csc(t) := 1/\sin(t)$ and $\cot(t) := 1/\tan(t)$, but frankly there is little benefit to have so many when $\cos,\sin$ alone suffice.) Whenever you wish to simplify any trigonometric expression involving $\cos,\sin,\tan,\sec,\csc,\cot$, you probably should perform the standard mathematical technique of rewriting in canonical form, which in this case means rewriting in terms of $\cos,\sin$ alone, while taking note of where the original expression is not defined (for instance, $1/\csc(t) = \sin(t)$ for any real $t$ only when $t$ is not a multiple of $π$).
The other key algebraic facts arise from considering rotation matrices applied to vectors. (If you are unfamiliar with matrices as operators on vectors, please read this first. For an introduction to vectors in euclidean space, see here.) Let $R$ be any rotation about the origin in the plane. Then $R$ satisfies three properties:
- $R(u+v) = R(u)+R(v)$ for any vectors $u,v$ (i.e. summing two vectors then rotating the result gives the same as rotating the two vectors first before summing them).
- If $R,S$ are anti-clockwise rotations of angles $t,u$ respectively, then $R∘S$ is an anti-clockwise rotation of angle $t+u$.
- If $R$ is an anti-clockwise rotation of angle $t$, then:
a. $R(⟨x,0⟩) = ⟨x·\cos(t),x·\sin(t)⟩$ for any real $x$.
b. $R(⟨0,y⟩) = ⟨-y·\sin(t),y·\cos(t)⟩$ for any real $y$.
We can take these properties as axioms (assumption) about rotations. After all, if $R$ does not satisfy them then we would not call $R$ a rotation to begin with. To see why, property (1) captures the intuition that rotating two connected rods will rotate both rods by the rotation angle while preserving where they connect. Property (2) is only needed in conjunction with property (3). Property (3a) follows from the definition of $\cos,\sin$, and property (3b) follows from the same definition rotated $90°$ anti-clockwise.
Properties (1) and (3) yield the matrix form of a 2d rotation:
If $R$ is an anti-clockwise rotation of angle $t$, then $R = \small \pmatrix{ \cos(t) & -\sin(t) \\ \sin(t) & \cos(t) }$.
And then using property (2) we get:
$\small \pmatrix{ \cos(t+u) & -\sin(t+u) \\ \sin(t+u) & \cos(t+u) } = \pmatrix{ \cos(t) & -\sin(t) \\ \sin(t) & \cos(t) } \pmatrix{ \cos(u) & -\sin(u) \\ \sin(u) & \cos(u) }$ for any reals $t,u$.
Multiplying the matrix product on the right and comparing with the matrix on the left immediately gives the angle-sum identities:
$\cos(t+u) = \cos(t)·\cos(u) - \sin(t)·\sin(u)$ for any reals $t,u$.
$\sin(t+u) = \cos(t)·\sin(u) + \sin(t)·\cos(u)$ for any reals $t,u$.
Whenever you wish to simplify expressions involving trigonometric functions on sums of angles, you should consider using these identities to reduce the expression to be in terms of $\cos,\sin$ of as few angles as possible.
In fact, all trigonometric identities involving only arithmetic operations and trigonometric functions can be proven using just the above definitions and key algebraic facts. A bit curiously, even the symmetry properties can be proven algebraically as follows.
Given any real $t$:
$1 = \cos(t+(-t)) = \cos(t)·\cos(-t) - \sin(t)·\sin(-t)$. [angle-sum]
$0 = \sin(t+(-t)) = \cos(t)·\sin(-t) + \sin(t)·\cos(-t)$. [angle-sum]
$\cos(t) = \cos(t)^2·\cos(-t) - (\cos(t)·\sin(-t))·\sin(t)$
$ = \cos(t)^2·\cos(-t) + (\sin(t)·\cos(-t))·\sin(t)$
$ = (\cos(t)^2+\sin(t)^2)·\cos(-t)$
$ = \cos(-t)$.
$\sin(t) = (\sin(t)·\cos(-t))·\cos(t) - \sin(t)^2·\sin(-t)$
$ = -(\cos(t)·\sin(-t))·\cos(t) - \sin(t)^2·\sin(-t)$
$ = -\sin(-t)·(\cos(t)^2+\sin(t)^2)$
$ = -\sin(-t)$.
Going on to real analysis, we would need the following facts, which can be taken as axioms for now (and justified separately later):
- $\sin' = \cos$.
- $\cos' = -\sin$.
As before, everything can be reduced to these, so there is no real need to memorize anything more (even though it may be convenient to do so).
Complex trigonometry
Personally, I think it is best to go straight to the complex-valued trigonometric functions, if one desires a complete and rigorous foundation for the mathematical field of analysis. One simply defines:
$
\def\rr{\mathbb{R}}
\def\cc{\mathbb{C}}
\def\lfrac#1#2{{\large\frac{#1}{#2}}}
$
$\exp(z) := \sum_{k=0}^∞ \lfrac{z^k}{k!}$ for every complex $z$ (after proving that the sum converges).
$\cos(z) := \lfrac{\exp(iz)+\exp(-iz)}{2}$.
$\sin(z) := \lfrac{\exp(iz)-\exp(-iz)}{2i}$.
$π$ is twice the first positive root of $\cos$ (after proving that it exists).
The motivation is that we want $\exp : \cc→\cc$ such that $\exp' = \exp$ and $\exp(0) = 1$, to be able to solve general linear differential equations, and we want $\cos,\sin : \rr→\rr$ such that $\cos'' = -\cos$ and $\sin'' = -\sin$ and $⟨\cos(0),\cos'(0)⟩ = ⟨1,0⟩$ and $⟨\sin(0),\sin'(0)⟩ = ⟨0,1⟩$, to be able to solve simple harmonic motion, and Taylor expansion brings us to the above definitions for $\exp,\cos,\sin$, which we can prove to converge on the entire complex plane. The above definition of $π$ is the easiest one that I know of that does not depend on any geometry. (For more details on this motivation, see this post.)
Suffice to say that with these definitions, we can prove by basic analysis that $\exp,\cos,\sin$ satisfy the desired motivating properties as well as another key property of $\exp$:
$\exp(z+w) = \exp(z)·\exp(w)$ for any complex $z,w$.
Using this property, we can prove all the trigonometric identities via algebraic manipulation alone (and they hold for complex variables and not just real variables).
For instance, given any complex $z$:
$\cos(z)^2+\sin(z)^2 = \lfrac{(\exp(iz)+\exp(-iz))^2}{4} - \lfrac{(\exp(iz)-\exp(-iz))^2}{4}$
$ = \exp(iz)·\exp(-iz) = \exp(0) = 1$.
Nevertheless, it is often still easier to first prove the same key algebraic facts for $\cos,\sin$ and then use them to prove other identities, than to reduce everything to $\exp$.
