Can you use sine and cosine in addition to tangent to find the magnitude of vector resultants?

Steps to Solve

1. Identify given vectors
   We have two vectors: a car traveling at $10 \text{ km/hr}$ to the right (east) and wind blowing at $2 \text{ km/hr}$ upward (north).

2. Set up the right triangle
   The car's velocity and the wind's velocity create a right triangle, where:

   • One leg (horizontal) is $10 \text{ km/hr}$ (car velocity).
   • The other leg (vertical) is $2 \text{ km/hr}$ (wind velocity).

3. Calculate the resultant vector using Pythagorean theorem
   To find the magnitude of the resultant vector ( R ): $$ R = \sqrt{(10^2 + 2^2)} $$

4. Calculate ( R )
   Simplifying the equation: $$ R = \sqrt{(100 + 4)} $$ $$ R = \sqrt{104} $$ $$ R \approx 10.198 \text{ km/hr} $$

5. Determine the direction using trigonometry
   Using the tangent function to find the angle ( \theta ): $$ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{2}{10} $$

6. Calculate ( \theta )
   Taking the arctan: $$ \theta = \tan^{-1}\left(\frac{2}{10}\right) $$ $$ \theta \approx 11.31^\circ $$

7. State the direction
   The direction of the resultant vector is $11.31^\circ$ north of east.