Precalculus / Unit 6: Applications of Trigonometry
6.6 Applications of Triangles and Vectors
Notes
- Draw a diagram and decide whether the Law of Sines or Cosines fits the given information.
- A resultant vector is the sum of vectors such as plane velocity plus wind velocity.
- Forces are in equilibrium when they add to the zero vector.
- Example: a plane flies east at 200 mph with a wind of 50 mph toward the south. The resultant is $\langle 200, -50 \rangle$, with speed $\displaystyle \sqrt{40000 + 2500} = 50\sqrt{17}$ mph.
Practice
A plane flies east at 200 mph while a wind blows south at 50 mph. What is its ground speed?
$\displaystyle \sqrt{200^2 + 50^2} = \sqrt{42500} = 50\sqrt{17}$.
From point $A$ to point $B$ is 100 m. Angle $A = 45^\circ$ and angle $B = 105^\circ$ in triangle $ABC$. What is the length of $BC$?
$C = 30^\circ$. By the Law of Sines, $\displaystyle BC = \frac{100\sin 45^\circ}{\sin 30^\circ} = 100\sqrt{2}$.
Two forces $\langle 5, 2 \rangle$ and $\mathbf{F}$ act on an object that stays at rest. What is $\mathbf{F}$?
The forces must sum to $\langle 0, 0 \rangle$.
Ready for more? Take the Unit 6 test.