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Solving Systems of Equations
This unit introduces how to systematically solve a system of equations, namely linear equations. Examples of non-linear systems, including systems of 3 unknowns will be of emphasis.
Graphs of Trigonometric Functions
The unit focuses primarily on how to graph periodic sinusoidal functions, and how to identify features of a waveform to produce an equation by inspection.
Polar Coordinate Functions
An introduction to the polar coordinate system.
Exponents and Radicals
This unit is an extension of what was introduced in Math 1131. To learn how to work with radicals, knowing your exponent laws in crucial. Hence, this unit begins with a thorough review.
This chapter introduces you to exponential functions, and how they can be solved using logarithms.
Trigonometric Identities and Equations
Graphing the Sine Function
Now that you know how to identify the amplitude, phase shift, and cycle when given a periodic sinusoidal function, it’s time you learn how they’re graphed via the steps outlined underneath. Be mindful that these steps are identical for sinusoidal functions containing cosine, with one exception in Step D – the cosine wave looks different than the sine wave. The function we’re graphing using the outline below is y = 2 sin (3x + 60°).
(a) Draw two horizontal lines, each at a distance equal to the amplitude a from the x axis.
Now mark the vertical axis with the amplitude, 2.
(b) Draw a vertical line at a distance from the origin equal to the period P. We now have a rectangle of width P and height 2×a.
(c) Subdivide the period P into four equal parts. Label the x axis at these points, and draw vertical lines through them.
(d) Lightly sketch in the sine curve.
If your equation contained cosine instead:
(e) Shift the curve by the amount of the phase shift.
Without going into too much detail about each step, let take a look at how to analyze and plot the three different equations below.
Keep in mind that the analysis of these equations is based-off of the general formula introduced in the previous section:
However, your equation might be written in a different style, where b is common factored within the trigonometric function: b(x + c/b), such as:
- If that’s the case, c represents the phase shift, so you don’t need to use the formula phase shift = –c / b anymore. Remember that a negative c value is a shift to the right, and vice versa. A few of these types of examples are shown at the every end of this lesson.
Equations of the form y = a·sin[b(x–c)] + d
Sometimes you may need to specify a few extra features about your waveform, such as the y-intercept, and the coordinates of your start and stop point of your sketched cycle. To find the y-intercept, you set x = 0. For example, using the function we started with:
- y = 2 sin (3x + 60°)
Setting x = 0:
- y = 2 sin (3(0) + 60°) ≈ 1.73
Therefore, the y-intercept point is (0, 1.73).
The start point for a sine wave begins at its center, then makes its way up to the maximum (known as the crest or peak), center, minimum (known as trough), then back to its center (end point). The cosine wave always starts at the wave’s maximum, passes through its center, reaches its minimum, center, then back to its maximum. Therefore, once you’ve graphed your wave, these points can be easily identified.