I was looking for examples that shows importance of trigonometry in an elementary level. Like if you wanted to surprise your high school students that how problems are made easy using trigonometry and if they didn't know about it the problem could be difficult (or very difficult) to solve!
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$\begingroup$I am not a teacher, but from a student's perspective, learning about how sine and cosine were approximated would help me find quite a bit more respect for trigonometric operations. Before I understood calculus, if I were shown a simple trig problem, and I solved it using one of the trig functions, and then my teacher showed me the Taylor series for the trig function I used (explaining that it is the closest approximation and that is what we would end up using if we didn't take advantage of the trigonometric relationships) I would be blown away. This might have the effect you are looking for, but this is only from my former perspective of "hard" math, your students may be different.
$\endgroup$ $\begingroup$Let them find a way to measure the height of a building without climbing it.
$\endgroup$ $\begingroup$A typical problem is that of engineers when they want to measure the height $T$ of an inaccessible object (tree, building, etc) when, for example, a river (in red below) intervenes . They measure the two alpha and beta angles and also the distance $a=AD$. Thus T is obtained easily by trigonometry eliminating the variable $x=DB$ which cannot be measured.
I think, beyond representation of complex numbers, or trigonometric equations (such as $sin²+cos²=1$), the fields which benefitted the most from trigonometry and for which there is no "easy fix" are cartography, astronomy, construction etc.
Erathostene measure of the circumference of the earth...
Distances between stars...
Pre-calculation of dimensions of buildings, structures etc...
$\endgroup$ $\begingroup$Representation of cartesian graphs that are very complex become very easy when trigonometry is applied. The ovals of Cassini, the Witch of Agnesi, the lituus, and the hypocycloid have simpler trigonometric equations compared to their cartesian representations.
E.g. The circle $x^2+y^2 = 25$ can be done on a graphing calculator by having $y_1 = +\sqrt{25-x^2}$ and $y_2 = -\sqrt{25-x^2}$. However, $y_1$ and $y_2$ have no real solutions when $x \gt 5$.
Converting the above to polar coordinates, using $x = r \cos \theta$ and $y = r \sin \theta$, plugging in and simplifying, we get the far more elegant and simple $r = 5.$
On the other hand, there are curves that trigonometry can't help. Converting the equation for the line $y = x$ to polar coordinates eliminates $r$ and results in the equation $\tan \theta = 1$, which represents the angle $\pi/4$. Conversely, attempting to get a closed form for Archemdes' spiral $r = \theta$ in Cartesian coordinates is not as neat; $x^2 + y^2 = (\arctan (y/x))^2$ is as far as we can get.
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