Regarding presenting trigonometry to students so that they see some reason for it:
In a sequence of courses we have developed as the W. M. Keck Curriculum Project the students first see sine and cosine in the context of writing 2X2 matrices for origin centered rotations. They then immediately see triangles created using connected scatter plots rotated when rotation matrices are used to guide the computation of the coordinates of the images of the vertices.
The complex number system is developed using simple transformations, including origin centered rotations. The motivation to do this is to try to make sense of square roots of negative real numbers which arise sometimes when using the quadratic formula.
Later, data is collected as a slinky toy is put in motion over a motion sensor. With a little care the frequence an amplitudes of the oscillations can be made rather constant. Here sine wave functions provide good models for the data.
We have found that navagation problems provide a good motivation for applying sine and cosine to triangles. Given the heading and distance for the first two legs of a journey, how far is one from the starting point and on what heading must one travel to return to the starting point?
We also apply triangle trigonometry to computing the distance between two points on a spherical model of the Earth given their longitude and latitude. This is done without bringing in the Spherical Law of Cosines with no explanation.
Paramatric simulations of projectile motion also naturally involved sine and cosine to describe separately the horizontal and vertical components of the motion.