Give an example, with justification:
A real life situation requiring the use of a trigonometry identity.
Knowledge is life, so open your doors of perception.
Here is one use of identities that is introduced to students in a high school physics class ... the derivation of the range formula for the horizontal displacement of a projectile (Galilean model).
start with the physics equation for vertical displacement ...
$\displaystyle \Delta y = v_0 \sin(\theta) t - \frac{1}{2}gt^2$
since the projectile lands at the same height it started,
$\displaystyle \Delta y = 0$ ...
$\displaystyle 0 = v_0 \sin(\theta) t - \frac{1}{2}gt^2$
solving for t ...
$\displaystyle t = \frac{2v_0 \sin(\theta)}{g}$
horizontal displacement is given by the formula ...
$\displaystyle \Delta x = v_0 \cos(\theta) t$
using the last two equations, eliminate the parameter t ...
$\displaystyle \Delta x = \frac{2v_0^2 \sin(\theta) \cos(\theta)}{g}$
using the double angle identity
$\displaystyle \sin(2\theta) = 2\sin(\theta)\cos(\theta)$,
the last equation becomes ...
$\displaystyle \Delta x = \frac{v_0^2 \sin(2\theta)}{g}$
from this equation, it can be deduced that the launch angle for maximum horizontal range is 45 degrees.