Since the Apollo 11 code is on GitHub, I was able to find the code that looks like an implementation of sine and cosine functions: see here for the command module and here for the lunar lander (it looks like it is the same code).
For convenience, here is a copy of the code:
# Page 1102
BLOCK 02
# SINGLE PRECISION SINE AND COSINE
COUNT* $$/INTER
SPCOS AD HALF # ARGUMENTS SCALED AT PI
SPSIN TS TEMK
TCF SPT
CS TEMK
SPT DOUBLE
TS TEMK
TCF POLLEY
XCH TEMK
INDEX TEMK
AD LIMITS
COM
AD TEMK
TS TEMK
TCF POLLEY
TCF ARG90
POLLEY EXTEND
MP TEMK
TS SQ
EXTEND
MP C5/2
AD C3/2
EXTEND
MP SQ
AD C1/2
EXTEND
MP TEMK
DDOUBL
TS TEMK
TC Q
ARG90 INDEX A
CS LIMITS
TC Q # RESULT SCALED AT 1.
The comment
# SINGLE PRECISION SINE AND COSINE
indicates, that the following is indeed an implementation of the sine and cosine functions.
Information about the type of assembler used, can be found on Wikipedia.
Partial explanation of the code:
The subroutine SPSIN actually calculates $\sin(\pi x)$, and SPCOS calculates $\cos(\pi x)$.
The subroutine SPCOS first adds one half to the input, and then proceeds to calculate the sine (this is valid because of $\cos(\pi x) = \sin(\pi (x+\tfrac12))$). The argument is doubled at the beginning of the SPT subroutine. That is why we now have to calculate $\sin(\tfrac\pi2 y)$ for $y=2x$.
The subroutine POLLEY calculates an almost Taylor polynomial approximation of $\sin(\tfrac\pi2 x)$. First, we store $x^2$ in the register SQ (where $x$ denotes the input). This is used to calculate the polynomial
$$
((( C_{5/2} x^2 ) + C_{3/2} ) x^2 + C_{1/2}) x.
$$
The values for the constants can be found in the same GitHub repository and are
$$\begin{aligned} C_{5/2} &= .0363551 \approx \big(\frac\pi2\big)^5 \cdot \frac1{2\cdot 5!}\\ C_{3/2} &= -.3216147 \approx -\big(\frac\pi2\big)^3 \cdot \frac1{2\cdot 3!}\\ C_{1/2} &= .7853134 \approx \frac\pi2 \cdot \frac12\\ \end{aligned}$$
which look like the first Taylor coefficients for the function $\frac12 \sin(\tfrac\pi2 x)$.
These values are not exact! So this is a polynomial approximation, which is very close to the Taylor approximation, but even better (see below, also thanks to @uhoh and @zch).
Finally, the result is doubled with the DDOUBL command, and the subroutine POLLEY returns an approximation to $\sin(\tfrac\pi2 x)$.
As for the scaling (first halve, then double, ...), @Christopher mentioned in the comments, that the 16-bit fixed-point number could only store values from -1 to +1. Therefore, scaling is necessary. See here for a source and further details on the data representation. Details for the assembler instructions can be found on the same page.
How accurate is this almost-Taylor approximation? Here you can see a plot on WolframAlpha for the sine, and it looks like a good approximation for $x$ from $-0.6$ to $+.6$. The cosine function and its approximation is plotted here. (I hope they never had to calculate the cosine for a value $\geq \tfrac\pi2$, because then the error would be unpleasantly large.)
@uhoh wrote some Python code, which compares the coefficients $C_{1/2}, C_{3/2}, C_{5/2}$ from the Apollo code with the Taylor coefficients and calculates the optimal coefficients (based on the maximal error for $-\tfrac\pi2 \leq x \leq \tfrac\pi2$ and quadratic error on that domain). It shows that the Apollo coefficients are closer to the optimal coefficients than the Taylor coefficients.
In this plot the differences between $\sin(\pi x)$ and the approximations (Apollo/Taylor) is displayed. One can see, that the Taylor approximation is much worse for $x\geq .3$, but much better for $x\leq .1$. Mathematically, this is not a huge surprise, because Taylor approximations are only locally defined, and therefore they are often only useful close to a single point (here $x=0$).
Note that for this polynomial approximation you only need four multiplications and two additions (MP and AD in the code). For the Apollo Guidance Computer, memory and CPU cycles were only available in small numbers.
There are some ways to increase accuracy and input range, which would have been available for them, but it would result in more code and more computation time. For example, exploiting symmetry and periodicity of sine and cosine, using the Taylor expansion for cosine, or simply adding more terms of the Taylor expansion would have improved the accuracy and would also have allowed for arbitrary large input values.
