Addition of Ordinates for Circular Functions Revision Notes for VCE SSCE Mathematical Methods

Addition of Ordinates for Circular Functions

Introduction

Sums of trigonometric functions are fundamental in mathematics and have numerous practical applications, including audio compression and signal processing. When we need to sketch the graph of a function that is the sum of two simpler functions, we use a technique called addition of ordinates.

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Addition of ordinates is a graphical method where we add the $y$ -values (ordinates) of two functions at various $x$ -values to obtain points on the graph of their sum.

Key principles for sketching sum functions

When sketching $y = (f + g)(x)$ , the following principles help us understand how the combined graph relates to the individual functions:

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When one function equals zero:

If $f(x) = 0$ at a particular point, then $(f + g)(x) = g(x)$ at that point. The sum function passes through the same point as $g(x)$ .
If $g(x) = 0$ at a particular point, then $(f + g)(x) = f(x)$ at that point. The sum function passes through the same point as $f(x)$ .

When both functions have the same sign:

If both $f(x)$ and $g(x)$ are positive, then $(f + g)(x) > g(x)$ and $(f + g)(x) > f(x)$ . The sum is greater than either individual function.
If both $f(x)$ and $g(x)$ are negative, then $(f + g)(x) < g(x)$ and $(f + g)(x) < f(x)$ . The sum is more negative than either individual function.

When functions have opposite signs:

If $f(x)$ is positive and $g(x)$ is negative, then $g(x) < (f + g)(x) < f(x)$ . The sum lies between the two functions.

Finding zeros:

Look for values of $x$ where $f(x) + g(x) = 0$ . These are the x-intercepts of the sum function.

Worked example

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Worked Example: Sketching Sum Functions

Problem: Using the same scale and axes, sketch the graphs of $y_1 = 2\sin x$ and $y_2 = 3\cos(2x)$ for $0 \leq x \leq 2\pi$ . Use addition of ordinates to sketch the graph of $y = 2\sin x + 3\cos(2x)$ .

Solution:

First, we sketch the two component functions $y_1 = 2\sin x$ and $y_2 = 3\cos(2x)$ on the same axes.

To construct the sum function $y = y_1 + y_2$ , we create a table of values at key points. We choose values of $x$ where the functions take on important values (maxima, minima, zeros).

The table shows:

At each $x$ -value:

We read the $y$ -value from $y_1 = 2\sin x$
We read the $y$ -value from $y_2 = 3\cos(2x)$
We add these values to get the $y$ -value for the sum function

For example:

At $x = 0$ : $y_1 = 0$ , $y_2 = 3$ , so $y = 0 + 3 = 3$
At $x = \frac{\pi}{4}$ : $y_1 = \sqrt{2}$ , $y_2 = 0$ , so $y = \sqrt{2}$
At $x = \frac{\pi}{2}$ : $y_1 = 2$ , $y_2 = -3$ , so $y = 2 + (-3) = -1$
At $x = \frac{3\pi}{2}$ : $y_1 = -2$ , $y_2 = -3$ , so $y = -2 + (-3) = -5$

The complete graph shows all three functions:

Notice how the blue curve (the sum) relates to the component functions according to our key principles. For instance, at $x = \frac{\pi}{4}$ where $y_2 = 0$ , the sum equals $y_1$ . At $x = \frac{3\pi}{2}$ where both functions are negative, the sum is more negative than either individual function.

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Key Points to Remember:

Addition of ordinates means adding the y-values of two functions at each $x$ -value to create their sum
When one function equals zero, the sum equals the other function
When both functions are positive, their sum is greater than either function alone
When both functions are negative, their sum is more negative than either function alone
When functions have opposite signs, the sum lies between them
Creating a table of values at key points helps sketch the sum function accurately

Addition of Ordinates for Circular Functions (VCE SSCE Mathematical Methods): Revision Notes