6. Two lighthouses A and B are 25km apart and A is due West of B. A submarine S is
on a bearing of 1370 from A and on a bearing of 1700 from B. Draw a picture
relevant to the situation, and find the distance of S from A and the distance of S from
B( give all your answers to three significant figures)
How do i solve this useing like sin rule or cosine?.
This question is trigonometry
Angle A = 137°-90°=47°
Angle B = 180°-70°=10°,
10°+90°= 100°
Angle S = 180°-47°-100°=33°
Sine rule
b/sin(Angle B)= s/sin(Angle S )
b/sin100=25/sin33
b= sin100*25/sin33
b= 45.2km, b=AS
a/sin(Angle A)= s/sin(Angle S)
a= sin47*25/sin33
a=33.6km, a=BS
AS=45.2km, BS=33.6km
Sorry Angle B is 180°-170°=10°
10°+90°=100°
To solve this problem using trigonometry, you can apply the Law of Cosines.
First, let's draw a picture to represent the situation:
B (25 km)
/
/
/
/ S
A /
Now, we can label the angles and the known side lengths:
- Angle A = 13.70 (submarine's bearing from point A)
- Angle B = 17.00 (submarine's bearing from point B)
- Side a = 25 km (distance between lighthouses A and B)
- Side c = x km (distance between submarine S and lighthouse A)
- Side b = y km (distance between submarine S and lighthouse B)
Now, using the Law of Cosines, we can calculate the unknown side lengths:
For lighthouse A:
cos(A) = (b^2 + c^2 - a^2) / (2bc)
Now, substituting the known values into the equation, we get:
cos(137°) = (y^2 + x^2 - 25^2) / (2y * x)
For lighthouse B:
cos(B) = (a^2 + c^2 - b^2) / (2ac)
Substituting the known values into the equation yields:
cos(170°) = (x^2 + y^2 - 25^2) / (2x * y)
Now, with the given values, we can solve for x and y.
After calculating x and y, the distance of S from A is x km, and the distance of S from B is y km, giving all answers to three significant figures.
To solve this problem using trigonometry, we can utilize the sine rule or the cosine rule. Let's go step by step to find the distances of S from A and B.
1. Draw a diagram: Start by drawing a diagram representing the given scenario. Draw two lighthouses A and B, with a distance of 25 km between them. Place the submarine S on a bearing of 1370 from A and on a bearing of 1700 from B. Label all the angles and sides relevant to the problem.
2. Apply the sine rule: We can use the sine rule to find the length of the side opposite the angle of interest. In this case, we can find the distance of S from A by using the following formula:
Distance of S from A / sin(angle at A) = Distance between A and B / sin(angle at B)
Substitute the given values into the equation:
Distance of S from A / sin(1370) = 25 / sin(1700)
Rearrange the equation to solve for the distance of S from A:
Distance of S from A = (25 * sin(1370)) / sin(1700)
Calculate the value, rounding to three significant figures.
3. Apply the cosine rule: We can use the cosine rule to find the length of the side adjacent to the angle of interest. In this case, we can find the distance of S from B by using the following formula:
Distance of S from B = sqrt(Distance of S from A)^2 + Distance between A and B^2 - 2 * (Distance of S from A) * Distance between A and B * cos(angle at A)
Substitute the known values into the equation:
Distance of S from B = sqrt((Distance of S from A)^2 + 25^2 - 2 * (Distance of S from A) * 25 * cos(1370))
Calculate the value, rounding to three significant figures.
By following these steps and using either the sine rule or the cosine rule, you can find the distances of S from A and B. Note that it's essential to ensure that the calculator is set to the correct angle measurement (degrees or radians) before performing the calculations.