Basic Trig Help

[newAD]

Hi guys,

I am seeking help for my project. I am kind of stuck here.

I am given 4 base station and a point on the Cartesian plane. How do I find the angles Θ1, Θ2, Θ3 & Θ4. Can someone advice me on how i can calculate them?

View attachment 2734
I really appreciate if someone could help me. Thanks

 

[HallsofIvy]

The simplest way to do it is to use the fact that the angle between two vectors, u and v, is given by \(\displaystyle cos(\theta)= \frac{u\cdot v}{|u||v|}\). That is, the cosine of the angle is given by the dot product of the two vectors divided by their lengths. Here, for example, "stor" is at (3, 7) while point1 is at (5, 10). the vector from "stor" to point1 is given by <5- 3, 10- 7>= <2, 3>. Point4 is (0, 5) so the vector from "stor" to point4 is given by <0- 3, 5- 7>= <-3, -2>. The dot product of those two vectors is (2)(-3)+ (3)(-2)= -12. The lengths of the two vectors is \(\displaystyle \sqrt{4+ 9}= \sqrt{13}\) and \(\displaystyle \sqrt{9+ 4}= \sqrt{13}\) also. So we have \(\displaystyle cos(\theta_1)= \frac{-12}{13}= -0.9231\) and \(\displaystyle \theta_1= arccos(-0.9231)= 157\) degrees, approximately.

 

[newAD]

The simplest way to do it is to use the fact that the angle between two vectors, u and v, is given by \(\displaystyle cos(\theta)= \frac{u\cdot v}{|u||v|}\). That is, the cosine of the angle is given by the dot product of the two vectors divided by their lengths. Here, for example, "stor" is at (3, 7) while point1 is at (5, 10). the vector from "stor" to point1 is given by <5- 3, 10- 7>= <2, 3>. Point4 is (0, 5) so the vector from "stor" to point4 is given by <0- 3, 5- 7>= <-3, -2>. The dot product of those two vectors is (2)(-3)+ (3)(-2)= -12. The lengths of the two vectors is \(\displaystyle \sqrt{4+ 9}= \sqrt{13}\) and \(\displaystyle \sqrt{9+ 4}= \sqrt{13}\) also. So we have \(\displaystyle cos(\theta_1)= \frac{-12}{13}= -0.9231\) and \(\displaystyle \theta_1= arccos(-0.9231)= 157\) degrees, approximately.

Thanks for the reply. You were spot on! Thanks!!

 

[newAD]

HINT:

Label the points:
A(5,10), B(10,5), C(5,0), D(0,5), U(3,7), V(3,0), W(0,7)

Look for right triangles; example:
see right triangle DUW?
DW = 2, UW = 3, hypotenuse DU = SQRT(2^2 + 3^2) = SQRT(13)
So you have lengths of all 3 sides: calculate angle DUW.

Now create point X(3,10); similarly calculate angle AUX.

Then angle AUD = 90 + angleDUW + angleDUX. (you should get ~157.38)

NOTE: what did you notice about angles DUW and DUX?
Repeat above for 2 more angles; the 4th angle will be 360 - (the 3 you found); OK?

If you can't follow the above, you're not ready for this problem.

Thanks for the reply. I tried this before posting and I was wondering does this method satisfy if the star(3, 7) is move to else where of the Cartesian plane?