A wildlife researcher is tracking a flock of geese. The geese fly 5.0 km due west, then turn toward the north by 30 ∘ and fly another 5.0 km .

Answer 1

You don't ask a question but I'll take a guess that you'd like to know how far the geese are from the researcher and in what direction.

Here is a sketch:

I'll define the first vector (due west) as V₁ and the second vector (north by 30°) as V₂.

Obviously the 2 vectors cannot be added as they are because they are in different directions. One way to add them is the break V₂ up into an x-component and a y-component. That way, we can add the x-component with V₁ because they will then both be in the same direction.

Using trig ratios with the second vector:

cos(30°) = V_2x/5 and simplifying: V_2x = 5cos(30°) = 4.33

sin(30°) = V_2y/5 and simplifying: V_2y = 5sin(30°) = 2.5

Combining V_2x with V₁ we get: 4.33 + 5 = 9.33 and, of course we have V_2y = 2.5 like this:

To find "d", use the Pythagorean Theorem: 9.33² + 2.5² = d² so d = 9.66 km

To find the angle θ, use inverse tangent: θ = tan^-1(2.5/9.33) = 15° north of due west

Another way to do this is to use the Law of Cosines on the first sketch I drew:

d² = 5² + 5² - 2(5)(5)cos(150°)

and then use the Law of Sines to find θ:

sin(150)/d = sin(θ)/5