Respuesta :
1. Triangle STU:
•
• Side m
• Side m
a) To find the length of side TU, we use the fact that is the ratio of the length of to , so .
So, would be m.
b) To calculate angle , we can use the cosine rule, which states , where , , and are the sides opposite to angles , , and respectively.
Taking the inverse cosine (arccos) of gives as approximately .
c) To calculate angle , we know that the sum of angles in a triangle is , so .
2. Triangle XYZ:
• Side cm
• Side cm
• Side cm
a) To find the size of angle , we can use the cosine rule again.
Taking the inverse cosine (arccos) of gives as .
b) To find the size of angle , we use the sine rule. The sine rule states , where , , and are the sides opposite to angles , , and respectively.
Taking the inverse sine (arcsin) of gives as approximately .
3. Boat Journey:
• Boat initially sails on a bearing of for km.
• Then it turns to and sails for km to reach Crow Island.
a) The length of the return journey would be the straight line distance from Crow Island back to Aardvark Island. This can be calculated using the Pythagorean theorem since we have a right triangle formed by the boat’s path. The distance would be km.
b) To find the bearing on which the pilot must steer his boat to return to Aardvark Island, we can use trigonometry. The angle formed between the boat’s initial path and the straight line back to Aardvark Island can be found using . Taking the arctan of gives the angle . Then, the bearing would be from the original bearing (since it’s the opposite direction).
4. Jason’s Walk:
• Jason walks initially on a bearing of for a distance meters.
• Then he walks twice as far on a new bearing of for a distance of meters.
Given that the total distance walked is meters, we can set up the equation to find , which is the distance he initially walked. Once we find , we can calculate the bearing required to return to his original position using trigonometry, considering the angles formed by his initial path and the new path.
•
• Side m
• Side m
a) To find the length of side TU, we use the fact that is the ratio of the length of to , so .
So, would be m.
b) To calculate angle , we can use the cosine rule, which states , where , , and are the sides opposite to angles , , and respectively.
Taking the inverse cosine (arccos) of gives as approximately .
c) To calculate angle , we know that the sum of angles in a triangle is , so .
2. Triangle XYZ:
• Side cm
• Side cm
• Side cm
a) To find the size of angle , we can use the cosine rule again.
Taking the inverse cosine (arccos) of gives as .
b) To find the size of angle , we use the sine rule. The sine rule states , where , , and are the sides opposite to angles , , and respectively.
Taking the inverse sine (arcsin) of gives as approximately .
3. Boat Journey:
• Boat initially sails on a bearing of for km.
• Then it turns to and sails for km to reach Crow Island.
a) The length of the return journey would be the straight line distance from Crow Island back to Aardvark Island. This can be calculated using the Pythagorean theorem since we have a right triangle formed by the boat’s path. The distance would be km.
b) To find the bearing on which the pilot must steer his boat to return to Aardvark Island, we can use trigonometry. The angle formed between the boat’s initial path and the straight line back to Aardvark Island can be found using . Taking the arctan of gives the angle . Then, the bearing would be from the original bearing (since it’s the opposite direction).
4. Jason’s Walk:
• Jason walks initially on a bearing of for a distance meters.
• Then he walks twice as far on a new bearing of for a distance of meters.
Given that the total distance walked is meters, we can set up the equation to find , which is the distance he initially walked. Once we find , we can calculate the bearing required to return to his original position using trigonometry, considering the angles formed by his initial path and the new path.