Trigonometry for physics (right triangles, components) Transitional Reasoning Download PDF

Questions

1. 1.

   During an engineering test, a technician pushes a heavy crate up a ramp inclined at an angle $\alpha$ to the horizontal by applying a purely horizontal force of magnitude $F$. To determine how this force influences motion along the incline, the technician resolves $F$ into components parallel ($F_{\parallel}$) and perpendicular ($F_{\perp}$) to the ramp surface. Based on right-triangle trigonometry, which pair of expressions correctly represents the magnitudes of these two components?

   • A) $F_{\parallel} = F \sin\alpha$ and $F_{\perp} = F \cos\alpha$
   • B) $F_{\parallel} = F \cos\alpha$ and $F_{\perp} = F \sin\alpha$
   • C) $F_{\parallel} = F \tan\alpha$ and $F_{\perp} = F \cos\alpha$
   • D) $F_{\parallel} = F \cos\alpha$ and $F_{\perp} = F \tan\alpha$
2. 2.

   An autonomous underwater glider travels along a straight path descending at an angle $\theta$ below the horizontal surface of the sea. An onboard acoustic sensor records that the glider maintains a constant horizontal velocity component of magnitude $v_x$. Based on right-triangle trigonometry, which pair of expressions correctly represents the glider's total speed $v$ and its vertical descent speed $v_y$?

   • A) $v = \frac{v_x}{\cos\theta}$ and $v_y = v_x \tan\theta$
   • B) $v = v_x \cos\theta$ and $v_y = v_x \sin\theta$
   • C) $v = \frac{v_x}{\sin\theta}$ and $v_y = \frac{v_x}{\tan\theta}$
   • D) $v = v_x \tan\theta$ and $v_y = \frac{v_x}{\cos\theta}$
3. 3.

   A rescue drone exerts a steady pulling force of magnitude $T$ on a tether attached to a ground crate. The tether is tilted such that it makes an angle $\theta$ with the vertical axis. Which expressions correctly determine the horizontal component ($T_x$) and vertical component ($T_y$) of this force?

   • A) $T_x = T \sin\theta$ and $T_y = T \cos\theta$
   • B) $T_x = T \cos\theta$ and $T_y = T \sin\theta$
   • C) $T_x = T \tan\theta$ and $T_y = T \cos\theta$
   • D) $T_x = T \sin\theta$ and $T_y = T / \cos\theta$