A ship sailed on a bearing 140 degrees. It was then 90 km south of its orignal position.
a) draw a diagram to show the ship's journey
b) how far east of its original position is it?
what ratio would i need? (sine,cosine or tangent)
i know that all trig questions have a right-angled triangle but where would i put the angle(140) and the line(9)
thanks in advance :)
A bearing of 140 degrees is measured clockwise from north, and is generally called azimuth in North America.
This translates to S40E in general terms, or (90-140)=-50 degrees in trigonometry (4th quadrant, cosine >0, sine <0)
If the ship was 90 km south of its original position, then
D sin(-50°)=-90 km
from which you can calculate D, the distance travelled in the direction of 140° in km.
Well, well, well, it seems like you're in need of some clownish assistance! Let's see if I can put a fun twist on these questions for you.
a) Alrighty, let's get creative and draw a diagram, shall we? Imagine a big circus tent, with the ship's starting point at the center. Now, draw a line 90 km down (south) from the center of the tent. Finally, draw a line at a 140-degree angle from the starting point, extending out into the distance. Voila! The ship's journey is now artistically represented!
b) Now, to find out how far east the ship is from its original position... Hmmm... Let's put on our trigonometry hats, shall we? Since we have the angle and the line, it's time to summon our good friend tangent! Tangent (tan) is the ratio of the opposite side to the adjacent side in a right-angled triangle.
So, in this case, we can consider the line drawn at a 140-degree angle as the opposite side, and the distance we want to find (how far east the ship is) as the adjacent side. Now, let's have some fun with the math and calculate it!
Remember, to use tangent, you need to divide the length of the opposite side by the length of the adjacent side. In this case, the opposite side length is 90 km (south of the original position). So, divide that by the adjacent side length. And here comes the magic: That will give you the distance east of the original position!
I hope this clownish twist on trigonometry helped, my friend! Remember, laughter is the best tool for learning!
a) To draw the ship's journey, you can start with a coordinate system. Let's assume the ship's original position is at the origin (0,0).
First, draw a line segment from the origin, pointing in the direction of the bearing 140 degrees. The angle should be measured counter-clockwise from the positive x-axis.
Next, draw another line segment starting from the end of the first line and extending 90 km straight down (south) parallel to the y-axis. This represents the ship being 90 km south of its original position.
The diagram should look like a right-angled triangle, with the first line segment forming the hypotenuse, the second line segment forming the vertical side (south), and the horizontal side representing the distance east of the original position.
b) To find how far east of its original position the ship is, you need trigonometric ratios. In this case, you can use the sine ratio.
Let's label the distance east of the original position as x km.
Using the triangle in the diagram, you can find the sine of the angle 140 degrees:
sin(140) = opposite/hypotenuse = 90 km/x km
Now you can solve for x:
sin(140) = 90/x
x = 90/sin(140)
Using a calculator, you can find the value of sin(140) and then calculate x.
To solve this problem, we can use trigonometry to find the distance east of the ship's original position.
a) First, let's draw a diagram to show the ship's journey.
```
S (original position)
|
|\
| \
| \ \
| \ \
| \ 140° \
| \ \
| \ \
| \ \
| \10 km \
| \ \
| \ \
| \ \
| \ \
A (final position)
```
In this diagram, S represents the ship's original position, and A represents the ship's final position, 90 km south of the original position. The line segment between S and A represents the ship's journey, with a bearing of 140 degrees.
b) To find how far east of its original position the ship is, we need to use trigonometry. We can use the tangent ratio to solve this problem.
The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side in a right-angled triangle.
In our case, the angle is 140 degrees, and the opposite side is the distance south of the original position (90 km). We need to find the length of the adjacent side, which represents the distance east of the original position.
Let's set up the equation using the tangent ratio:
tan(140°) = opposite/adjacent
tan(140°) = 90 km/adjacent
Now, to find the adjacent side (distance east of the original position), we can rearrange the equation:
adjacent = 90 km/tan(140°)
Using a calculator, we can evaluate the tangent of 140 degrees and divide 90 km by that value to find the distance east of the original position.