due east is 90º
110º is 20º south of east
distance south of the start is ... 2 km * sin(20º)
a hiker travels 3 km due east then 2km on a bearing of 110° how far south is the hiker from the starting point
3 answers
Draw a horizontal line.
Mark start point as A.
Mark the point after 3 km as B.
At that point B, draw an angle θ = 110 ° with respect to the horizontal.
Mark the point after 2 km as C.
∠ θ = 110° = ∠ ABC
∠ is the mark for angle
Mark the length AB as a.
a = 3 km
Mark the length BC as b.
b = 2 km
Mark the length AC as c.
Now law of cosines:
c² = a² + b² - 2 ∙ a ∙ b ∙ cos θ
c² = a² + b² - 2 ∙ a ∙ b ∙ cos 110°
c² = 3² + 2² - 2 ∙ 3 ∙ 2 ∙ cos 110°
c² = 9 + 4 - 12 ∙ cos 110°
c² = 13 - 12 ∙ ( - 0.342020143 )
c² = 13 + 4.104241716
c² = 17.104241716
c = √17.104241716
c = 4.1357274712 km
Mark start point as A.
Mark the point after 3 km as B.
At that point B, draw an angle θ = 110 ° with respect to the horizontal.
Mark the point after 2 km as C.
∠ θ = 110° = ∠ ABC
∠ is the mark for angle
Mark the length AB as a.
a = 3 km
Mark the length BC as b.
b = 2 km
Mark the length AC as c.
Now law of cosines:
c² = a² + b² - 2 ∙ a ∙ b ∙ cos θ
c² = a² + b² - 2 ∙ a ∙ b ∙ cos 110°
c² = 3² + 2² - 2 ∙ 3 ∙ 2 ∙ cos 110°
c² = 9 + 4 - 12 ∙ cos 110°
c² = 13 - 12 ∙ ( - 0.342020143 )
c² = 13 + 4.104241716
c² = 17.104241716
c = √17.104241716
c = 4.1357274712 km
"how far south is the hiker from the starting point"
means that you only want the y co-ordinate of the resulting vector
(3cos0 , 3sin0) + (2cos(-20), 2sin(-20))
= (3,0) + (1.8794, -.684)
= (4.8794, -.684)
So the southward position from his starting point is .684 km
If we want the distance from his starting points we get
√(4.8794^2 + (-.684)^2) = 4.927 km
If you change the angle in Bosnian's solution to 160°, we get the same distance answer.
means that you only want the y co-ordinate of the resulting vector
(3cos0 , 3sin0) + (2cos(-20), 2sin(-20))
= (3,0) + (1.8794, -.684)
= (4.8794, -.684)
So the southward position from his starting point is .684 km
If we want the distance from his starting points we get
√(4.8794^2 + (-.684)^2) = 4.927 km
If you change the angle in Bosnian's solution to 160°, we get the same distance answer.