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:
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Measure the Height:
- Use a measuring tape to measure the height (h) of your friend or family member. Record this height in meters.
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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.
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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.
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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.
- If the height (h) is 1.7 meters and the length of the shadow (s) is 2 meters:
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:
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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°)
- Quadrants are defined as follows:
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Finding r:
- The distance (r) from the origin to the point P(x, y) can be calculated using the Pythagorean theorem: \[ r = \sqrt{x^2 + y^2} \]
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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: \[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
- Given the format P(r, 3), we assume you want to express the point in polar coordinates:
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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.