New Model Multipurpose High School

Annual Continuous Assessment Examination - 2024

ninth class

Investigative work: 01

1. Measure the height of your friend or family member to determine the elevation angle of the sun at any time of the day and determine the length of the shadow at that time.

Assignment 01

2. Take any point P (x, y) using two-dimensional coordinates such that adding the origin to it produces the quadrant angle. Find the trigonometric projection of the point at the ideal position with respect to the positive direction of the x-axis and express the point by P (r, 3).

1 answer

Let's break down the assignment into two parts: measuring the height and determining the shadow length, and then finding the trigonometric projection for the point P(x, y).

Part 1: Measuring Height and Shadow Length

Objective: Measure the height of a friend or family member to determine the elevation angle of the sun and calculate the length of the shadow.

Steps to Follow:

  1. Measure the Height:

    • Use a measuring tape to measure the height (h) of your friend or family member. Record this height in meters.
  2. Determine the Shadow Length (s):

    • At a certain time of day, measure the length of the shadow when your friend or family member stands upright. Record this length in meters.
  3. Calculate the Elevation Angle (θ):

    • You can use the tangent function, which relates the height and shadow length: \[ \tan(\theta) = \frac{h}{s} \]
    • Rearranging this gives: \[ \theta = \tan^{-1}\left(\frac{h}{s}\right) \]
    • Using a calculator or a trigonometric table, compute the angle θ in degrees.
  4. Example Calculation:

    • If the height (h) is 1.7 meters and the length of the shadow (s) is 2 meters:
      • Calculate the elevation angle: \[ \theta = \tan^{-1}\left(\frac{1.7}{2}\right) \approx \tan^{-1}(0.85) \approx 40.4^\circ \]
    • Record your findings.

Part 2: Trigonometric Projection of Point P(x, y)

Given that point P is located such that when it is added to the origin, it reflects the quadrant angle, let's denote this point as P(x, y).

Steps to Find Trigonometric Projection:

  1. Understanding Quadrants:

    • Quadrants are defined as follows:
      • Quadrant I: Both x and y are positive (0° to 90°)
      • Quadrant II: x is negative, y is positive (90° to 180°)
      • Quadrant III: Both x and y are negative (180° to 270°)
      • Quadrant IV: x is positive, y is negative (270° to 360°)
  2. Finding r:

    • The distance (r) from the origin to the point P(x, y) can be calculated using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} \]
  3. Expressing Point P:

    • Given the format P(r, 3), we assume you want to express the point in polar coordinates:
      • The angle θ with respect to the positive x-axis can also be found using: \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
  4. Example:

    • If P(3, 4): \[ r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
    • Thus, the point in polar coordinates would be P(5, θ) where: \[ θ = \tan^{-1}\left(\frac{4}{3}\right) \approx 53.13^\circ \]
    • This expression gives the radius and angle from the x-axis.

Now you can proceed with both parts of your assignment! Make sure to document your measurements and calculations clearly.