Lesson Notes By Weeks and Term v4 - SHS 3
MEASUREMENT
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MEASUREMENT
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Subject: Mathematics
Class: SHS 3
Term: 2nd Term
Week: 11
Grade code: 3.3.2.LI.2
Strand code: 3
Sub-strand code: 2
Content standard code: 3.3.2.CS.1
Indicator code: 3.3.2.LI.2
Theme: GEOMETRY AROUND US
Subtheme: MEASUREMENT
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As-salaam alaikum / Good morning, class. Look at the wavy pattern on this piece of Kente cloth, or think about the rhythm of Adowa music. Many things in our world move in cycles or waves. The rise and fall of the tide at Labadi beach, the changing seasons from harmattan to rainy season, and even the electricity supplied by ECG all follow a repeating pattern. Mathematics gives us a powerful tool to describe these patterns: trigonometric functions like sine and cosine. Today, we are going to learn how to be "detectives." We will look at the "picture" of a wave (its graph) and figure out the exact mathematical equation that created it.
(30 minutes)
Today we will focus on finding the equation of a trigonometric graph in the general form: `y = A sin(B(x - C)) + D` or `y = A cos(B(x - C)) + D`
Let's break down what each letter means. Think of them as control knobs that change the shape of the basic `sin(x)` or `cos(x)` graph.
The Four Key Parameters: The Vertical Shift (D) - The Middle Line This is the horizontal line that runs exactly in the middle of the wave, halfway between the highest point (maximum) and the lowest point (minimum). It tells us how much the graph has been shifted up or down from the x-axis. How to find it: Find the maximum value (Max) and the minimum value (Min) of the graph. Calculate the average: `D = (Max + Min) / 2` The Amplitude (A) - The Height of the Wave This is the distance from the middle line (D) to the maximum point, or from the middle line to the minimum point. It's always a positive value. It tells us how "tall" or "intense" the wave is. How to find it: `A = Max - D` or `A = D - Min` A simpler formula is: `A = (Max - Min) / 2` Note: If the graph starts by going down from the midline (for sine) or starts at a minimum (for cosine), we use a negative sign for `A` in the final equation (e.g., `y = -2 sin(x)`). The value of `A` itself is the distance, which is positive. The Period and the value of B - The Length of a Cycle The Period is the length along the x-axis for one full wave or cycle before it starts repeating. `B` is a value in the equation that determines the period. It tells us how many cycles fit into the standard 360° (or 2π radians). How to find the Period: Pick a convenient starting point (like a maximum). Find the x-value of the next maximum. The difference is the period. Or, measure the horizontal distance from one point on the wave to the next identical point where the wave is moving in the same direction. How to find B from the Period: Once you know the period, use the formula: `B = 360° / Period` (if x-axis is in degrees) or `B = 2π / Period` (if x-axis is in radians). We will mostly use degrees today. The Phase Shift (C) - The Starting Point This is the horizontal shift of the graph. It tells us how far the graph has been moved to the right or left. How to find it (choose one method): For a sine graph (`y = A sin(...)`): A standard sine graph starts at the midline (`y=D`) and goes up. Find the x-coordinate of the first point where your graph crosses the midline and is increasing. This x-coordinate is `C`. For a cosine graph (`y = A cos(...)`): A standard cosine graph starts at its maximum value. Find the x-coordinate of a maximum point. This x-coordinate is `C`. Important: The value `C` is the shift. In the formula `(x - C)`, if the graph shifts right by 30°, `C = 30`, and the bracket is `(x - 30)`. If it shifts left by 30°, `C = -30`, and the bracket becomes `(x - (-30))` which is `(x + 30)`.