# Error analysis / propagation : trigonometry

If the hypotenuse of a triangle is (1536 +- 3)m long, and the (non right-angle) angle measured from the ground is (22.2 +-0.1) degrees, what is the height of the triangle, and the error in this?

sin(theta) = opp/hypotenuse = h/x

h = x sin(theta)

dh = ?

• Did you try googling error propagation rules? First result has the answer.... – lemon May 13 '16 at 14:19

Assume the values are normally distributed without correlation (i.e. $\Delta(x+y) = \sqrt{\Delta x^2 + \Delta y^2}$), then
\begin{align} \Delta h &= \Delta ( x \sin \theta ) \\ &= \sqrt{ (\Delta x \cdot \sin \theta)^2 + (x \cdot \Delta (\sin \theta))^2 } \tag{product rule} \\ &= \sqrt{ (\Delta x \cdot \sin \theta)^2 + (x\cos \theta \cdot \Delta \theta)^2 }. \tag{$\tfrac d{d\phi}\sin \phi = \cos \phi$} \end{align}