# Simple Trig (1 Viewer)

#### nrlwinner

##### Member
Can someone help me because I just can't seem to get this.

A wooden stake, S, is 13m from a point, A, on a straight fence. SA makes an angle of 20degrees with the fence. If a goat is tethered to S by a 10m rope, where, on the fence, is the nearest point to A at which it can graze.

#### Dumbledore

##### Member
i can't really draw the diagram here and i'm not sure if i even got it right but

it looks like you have to use the cosine rule to find the distance from point A

u end up getting: 100 = x^2 + 169 - (26cos20)x but this quadratic is messy
solving this quadratic u get x ~= 3.26, u will get 2 solutions because there are two possible ways for the triangle to work, u take the smaller one since u want it closer to A

Alternatively

use the sine rule to find the angle between rope and fence,
u get @=26.4, 180-26.4(153.6)[since sine is + in first 2 quadrants and it can be either since they are both < 180]
next since it says u want the closest value to the fence, u use the optuse angle(153.6) and use the sum of triangle to figure out the angle between SA and the rope, u get 180-153.6-20 = 6.4 then use the sine rule once more and get x ~=3.26

i think ur supposed to use the 2nd method but both seem to get the same answer

#### nrlwinner

##### Member
i can't really draw the diagram here and i'm not sure if i even got it right but

it looks like you have to use the cosine rule to find the distance from point A

u end up getting: 100 = x^2 + 169 - (26cos20)x but this quadratic is messy
solving this quadratic u get x ~= 3.26, u will get 2 solutions because there are two possible ways for the triangle to work, u take the smaller one since u want it closer to A

Alternatively

use the sine rule to find the angle between rope and fence,
u get @=26.4, 180-26.4(153.6)[since sine is + in first 2 quadrants and it can be either since they are both < 180]
next since it says u want the closest value to the fence, u use the optuse angle(153.6) and use the sum of triangle to figure out the angle between SA and the rope, u get 180-153.6-20 = 6.4 then use the sine rule once more and get x ~=3.26

i think ur supposed to use the 2nd method but both seem to get the same answer
Yes you are right. Can someone put up a diagram because I can't get the what the actual question is saying.

#### jet

##### Banned

$\bg_white \text{The goat can graze at 2 points on the fence } G \text{ and } G'. \\ \text{Using the cosine rule } GS^2 = AS^2 + AG^2 - 2AS \cdot AG cos \, \angle SAG \\ 100 = 169 + x^2 - 26xcos20 \\ x^2 - \left (26cos \, 20^{\circ} \right )x + 69 = 0 \\ \text{The point } G \text{ will have the smaller value of } x \text{, which will also be the answer} \\ x = \frac{26cos \, 20^{\circ} - \sqrt{676cos^2 \, 20^{\circ} - 276}}{2} \\ = 13cos \, 20^{\circ} - \sqrt{169cos^2 \, 20^{\circ} - 69} \\ = 3.26 \text{ m}$

#### Galactic Drama

##### New Member
A bit late but where did you get this question from?

#### Drongoski

##### Well-Known Member
A bit late but where did you get this question from?
The poster most likely no longer active on Bored.

#### HazzRat

##### Active Member
A bit late but where did you get this question from?
Last post July 2009,..

#### tywebb

##### dangerman
A bit late but where did you get this question from?
It is from New Senior Mathematics Two Unit Course for Years 11 & 12 by Fitzpatrick. (1st ed, 1984) Ex 8d Q10 page 191

However it has been reused in the second and 3rd editions too.

In the more up-to-date 3rd edition it is Ex 2.8 Q10 on page 47 (slightly reworded, but essentially the same).

#### Galactic Drama

##### New Member
It is from New Senior Mathematics Two Unit Course for Years 11 & 12 by Fitzpatrick. (1st ed, 1984) Ex 8d Q10 page 191

However it has been reused in the second and 3rd editions too.

In the more up-to-date 3rd edition it is Ex 2.8 Q10 on page 47 (slightly reworded, but essentially the same).
Thankyou