You are required to complete all the 4 tasks in this assignment. You can use scientific calculators or online scientific calculators for this assignment. When you are instructed to make a graph in this assignment, please use GeoGebra graphing tool. Show all steps wherever necessary. Task 1:After completing one revolution, point A is observed to have coordinates (() on the unit circle. Using that information, please answer the following questions:(i) Calculate all 6 values of the trigonometric functions (clearly state the formulae used to calculate these functions).(ii) Determine the quadrant in which the point lies and provide the reason.(iii) Calculate the angle formed by point A and the reference angle with the positive X-axis.Task2: Alice had been standing on the ground (Point A) and observing a brightly colored object resembling a bird on the top of a tree at a distance of 40 meters from the tree. She decided to get a closer look by moving 20 meters closer to the tree (Point B). After moving closer, she realized that the object was not a bird but something that she could catch. Then, she decided to catch it by climbing the tree, which had a height of 60 meters from the ground. Using the above scenario, please answer the following questions showing step by step calculations and stating the formulae.(i) Find the angles formed by Alice at the points A and B relative to the top of the tree. What are these angles called as?(ii) Determine whether angle A is larger than angle B. Make a conclusion about the comparison of angles when observing an object from a distance versus close.(iii) Find the distances between the object and points A and B.
Question
You are required to complete all the 4 tasks in this assignment. You can use scientific calculators or online scientific calculators for this assignment. When you are instructed to make a graph in this assignment, please use GeoGebra graphing tool. Show all steps wherever necessary. Task 1:After completing one revolution, point A is observed to have coordinates (() on the unit circle. Using that information, please answer the following questions:(i) Calculate all 6 values of the trigonometric functions (clearly state the formulae used to calculate these functions).(ii) Determine the quadrant in which the point lies and provide the reason.(iii) Calculate the angle formed by point A and the reference angle with the positive X-axis.Task2: Alice had been standing on the ground (Point A) and observing a brightly colored object resembling a bird on the top of a tree at a distance of 40 meters from the tree. She decided to get a closer look by moving 20 meters closer to the tree (Point B). After moving closer, she realized that the object was not a bird but something that she could catch. Then, she decided to catch it by climbing the tree, which had a height of 60 meters from the ground. Using the above scenario, please answer the following questions showing step by step calculations and stating the formulae.(i) Find the angles formed by Alice at the points A and B relative to the top of the tree. What are these angles called as?(ii) Determine whether angle A is larger than angle B. Make a conclusion about the comparison of angles when observing an object from a distance versus close.(iii) Find the distances between the object and points A and B.
Solution
Task 1: (i) The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). The formulas for these functions are as follows:
sin = opposite/hypotenuse cos = adjacent/hypotenuse tan = opposite/adjacent csc = 1/sin sec = 1/cos cot = 1/tan
However, without the coordinates of point A, we cannot calculate the values of these functions.
(ii) The quadrant in which point A lies cannot be determined without the coordinates of point A.
(iii) The angle formed by point A and the reference angle with the positive X-axis cannot be calculated without the coordinates of point A.
Task 2: (i) The angles formed by Alice at points A and B relative to the top of the tree can be calculated using the tangent function (tan) of trigonometry. The formula is tan(theta) = opposite/adjacent.
At point A: tan(theta) = height of tree/distance from tree = 60/40 theta = arctan(60/40)
At point B: tan(theta) = height of tree/distance from tree = 60/20 theta = arctan(60/20)
These angles are called the angles of elevation.
(ii) By comparing the values of theta at points A and B, we can determine whether angle A is larger than angle B. Generally, as the distance from the object decreases, the angle of elevation increases. Therefore, angle B is likely larger than angle A.
(iii) The distances between the object and points A and B can be calculated using the Pythagorean theorem, which states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The formulas are as follows:
Distance from A to object = sqrt((distance from A to tree)^2 + (height of tree)^2) = sqrt((40)^2 + (60)^2)
Distance from B to object = sqrt((distance from B to tree)^2 + (height of tree)^2) = sqrt((20)^2 + (60)^2)
Similar Questions
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