7.5
三角関数：問題解決
ロケットの打ち上げなど、固定された地上の位置から物体が垂直に上昇する様子を観測する場合、三角関数の関係を利用すると、その物体の高さを正確に求めることができます。物体が上昇していくにつれて、打ち上げ地点から既知の水平距離にいる観測者は、地面と物体への視線がなす仰角を測定します。この仰角の変化は、観測さ…
人はロケットから一定の距離に立って、垂直打ち上げの準備をしています。
ロケットが上方に移動すると、その位置と仰角は飛行中に連続的に変化します。
三角関数は、この変化する角度をロケットの垂直高さ、絶対距離、および地面距離に結び付けます。
接線関数は、ロケットの垂直高さを観測角度と固定された地面距離に関連付けます。
ロケットの高さは、既知の地上距離に測定された角度の接線を掛けることによって求められます。
角度の正弦はロケットの垂直高さと絶対距離の比を示し、余弦は地上距離と絶対距離の比を示します。
垂直方向の高さがわかると、正弦波は高さを使用して絶対距離を計算でき、余弦波は地上距離を使用して同じことを行うことができます。
角度が大きくなると、これらの三角関数関係は、計算された高さとロケットまでの観測距離の両方に影響します。
これらの関数を適用することで、観測者は測定角度からのロケットの高さ、絶対距離、および地面距離を三角測量できます。
View the full transcript and gain access to JoVE Core videos
Q1: How does the tangent function help calculate a rocket's height during launch?
The tangent function relates the rocket's vertical height to the fixed ground distance and the measured angle of elevation. To find height, multiply the known ground distance by the tangent of the angle between the ground and the observer's line of sight. This direct relationship makes tangent the primary tool for height calculations in rocket observation scenarios.
Q2: What role do sine and cosine play in determining rocket position?
Sine represents the ratio of the rocket's vertical height to the absolute distance along the observer's line of sight, while cosine represents the ratio of ground distance to that same slanted distance. Although not used directly for height calculation, these trigonometric functions describe the geometric proportions of the right triangle formed by the ground, vertical height, and line of sight.
Q3: How can you find the absolute distance to a rocket using trigonometric functions?
Once the rocket's vertical height is known, you can calculate absolute distance using either sine or cosine. Sine uses the height and the angle to find the slanted distance, while cosine uses the ground distance and angle. Both methods yield the same absolute distance along the observer's line of sight to the rocket.
Q4: Why does the angle of elevation change as a rocket ascends?
As the rocket rises vertically, its position relative to the observer changes continuously, causing the angle between the ground and the line of sight to increase. This dynamic angle reflects the rocket's changing height and distance, providing real-time information about its position throughout the flight.
Q5: What information must an observer know to use trigonometric functions for rocket tracking?
An observer must know the fixed horizontal distance from their position to the launch site and measure the angle of elevation to the rocket at any given moment. With these two pieces of information, trigonometric functions can determine the rocket's vertical height, absolute distance, and ground distance with precision.
Q6: How do trigonometric functions change as the angle of elevation increases?
As the angle of elevation increases during the rocket's ascent, the values of tangent, sine, and cosine change in predictable ways according to trigonometric functions of real numbers. These changing values directly affect both the calculated height and observed distance, providing a mathematical framework to track vertical position over time.
Q7: How does triangulation using trigonometry determine multiple rocket measurements simultaneously?
By applying tangent, sine, and cosine functions to the measured angle and known ground distance, observers can simultaneously calculate the rocket's height, absolute distance, and ground distance. This triangulation method uses the right triangle formed by the observer, launch point, and rocket position to extract all three measurements from a single angle measurement.