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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
Introduction to Periodic Functions
In Part 1 of this course (Math 1131), you were newly introduced to the trigonometric functions: sine, cosine, and tangent. You learned how you can use these functions to solve triangles by setting up ratios, but you never learned what they looked like graphed. It turns out that if you select angles to represent θ for, let’s say y = sin(θ) or y = cos(θ), starting from θ = 0°, a repeating wave it formed. In fact, after every ±360°, the wave cycle repeats itself. Hence, trigonometric functions are commonly used to represent periodic functions – equations, that when graphed, repeat themselves indefinitely unless limits or bounds are defined. When the periodic function produces smooth symmetrical waves, where any portion of the wave can be horizontally translated onto another portion of the curve, it is referred to as a sinusoidal function.
Anything that repeats itself over-and-over can be represented using a periodic function, this includes pistons moving up-and-down inside a car engine, a Ferris wheel, sound waves, television signals, etc. The algorithms to all these can be represented mathematically using periodic functions (sine, cosine, tangent) – this is one way we use math to quantify and make sense of the world with numbers.
Our main focus in this chapter will be the sine and cosine function, which has wide applications to alternating current (electricity), mechanical vibrations, and so forth. Our task in this chapter will be to graph such functions containing either sine or cosine, building upon our earlier methods for graphing, and to extract useful information from the function.
Sine Function Analysis
Recall that sine is a ratio comparing the length of the opposite side of a right triangle to the length of its hypotenuse. If we keep the hypotenuse constant at a length of 1, at each given angle starting from an angle of 0°, the ratio will be different. A completed table showing the outputs of sine at angles from 0° to 360° as shown below (intervals of 30° were used for simplicity sake).
If you plot the angles along the x-axis and the outputs along the y-axis, you should get a waveform that looks like this:
Had you continued from 360° to 720°, the wave would have repeated itself. Each repeated portion of the curve is called a cycle. Therefore, in the wave above, you see only 1 complete cycle. The frequency is the number of cycles it completes in a given interval (usually the period). The period of any periodic waveform is the horizontal distance occupied by one cycle.
Summary: For the waveform above:
Number of cycles displayed: 1
Period: 360° / 1 cycle
Frequency: cycles / period = 1 cycle per 360°
Depending on the units used to represent the horizontal axis, the period is commonly expressed as:
- degrees per cycle
- radians per cycle
- seconds per cycle
- or any units per cycle
The back-and-forth movement of a waveform is referred to as oscillation. Notice how y = sin(θ) oscillates between 1 and -1 along the y-axis. In nature, however, not everything oscillates between a -1 and 1. Similarly, not everything starts at x = 0° and ends at 360° either. In addition, the period might be shorter than 360 degrees / cycle, thereby creating several cycles per 360° or longer than 360°.
Whenever there is a difference in the wave’s amplitude (also referred to as the height), a difference in period, or a difference from where the cycle begins (horizontal offset), this is called a transformation of the period function.
The first transformation we’ll focus on is amplitude. The amplitude refers to the distance from the wave’s center to its peak.
To modify the amplitude of a waveform, a factor a (representing any real number) in front of the period function is multiplied with the trigonometric function.
Remember that originally, our equation was y = 1 sin(θ), where the amplitude equaled to 1. This is why the wave oscillated between -1 and +1. Had our equation been, say y = 1.5 sin(θ), then the wave would have oscillated between -1.5 and +1.5 (amplitude being 1.5). The relationship between factor a and amplitude is summarized below.
Notice that whatever your a value is, you always take its absolute value to quantify the wave’s amplitude. Hence, if your equation was y = -3 sin(θ), the amplitude would be a = 3 because the absolute of -3 → |-3| is positive 3.
- Therefore, if you every report a negative amplitude, it is wrong.
Test yourself: Given the sine waves shown below, state the equations for each given what you just learned about amplitude:Solutions
Red: y = 2 sin (θ)
Black: y = 1 sin (θ)
Fuschia: y = 1.5 sin (θ)
Green: y = -2 sin (θ) → more on this below[collapse]
Notice how when the amplitude is greater than one, the wave gets taller (or skinnier). This is why modifying the amplitude is often referred to as a vertical stretch or vertical compression, depending on the a value.
- If a > 1 or a < -1 → vertical stretch
- If -1 < a < 1 → vertical compression
When the leading factor a is negative, it causes the wave to reflect about its center. The best way to show that the wave is reflected when graphing (next section) is to reverse the signs of the wave’s peaks. In other words, locate the maximum and minimum points within a cycle, then change the y-coordinates from positive to negative or negative to positive, as shown below:
Cycle, Period, and Frequency
To manipulate the period of a waveform, that is, to make a cycle shorter or longer, factor b needs to change.
Let’s see what happens when you change b from 1, which is what it was originally in y = sin(1θ), to 2 or 0.5:
Since two cycles were completed in y = sin(2θ) within the same period as y = sin(θ), and only half a cycle was completed in y = sin(0.5θ), we can conclude that:
- When b is between -1 and 0 or 0 and 1, the period per cycle increases − gets bigger relative to 360º.
- -1 < b < 1 (where b ≠ 0), period increases.
- When b is less than -1 or greater than 1, the period per cycle decreases − gets smaller relative to 360º.
- b > 1 or b < -1, period decreases.
- Therefore, the factor b represents the functions cycle.
The relation between the cycle and period can be summarized as:
Remember that both of these equations can be manipulated to isolate for b if you’ve been given the graph to a wave, from which you can locate the period along the horizontal axis to then solve for b.
Because frequency represent cycles per period, the frequency can be found by taking the reciprocal of the period or:
Question: Find the period and frequency of the function y = sin(6x) both in degrees and radians, and graph one cycle.
From the equation, it’s clear that b = 6. We know that when b > 1, the period gets smaller. In fact, 6 waves can fit in the span of 1 wave cycle whose period is 360° or 2π rad (shown below).
Remember, not all cycles start at x = 0; some might before or after. To shift a periodic function to the left of the origin, a value must be added to the angle (or whatever the variable represents), and to shift it to the right, a value must be subtracted from the variable. Another term for phase shift common found in literature is horizontal translation. For instance:
- y = sin(x + 45°) shifts the wave to the left by 45° (see blue wave)
- y = sin(x – 45°) shifts the wave to the right by 45°. (see green wave)
- Generally, this value is denoted by the letter c [ y = a·sin(bθ + c) ], and is related to the phase-shift by the formula:
- In the examples provided above, b was equal to 1, so the formula technically wasn’t needed. Therefore, you use the formula when the periodic function contains a b value other than 1.
Another interesting feature about the value c is that it’s usually an excellent indicator of whether the equation written in degrees or radians. As illustrated above, c is in degrees, therefore you’d use the formula 360/b instead of 2π/b to find the period.
While there’s no focus on vertical shifts in this course, it’s worth to mention how it happens. If you want to shift the wave up or down, ±d is applied to the equation: y = a·sin(bθ + c) + d. A positive d value shifts it up d units, and a negative d value shifts it down d units. Another term for vertical shift common found in literature is vertical translation upwards or downwards.
Putting it all Together
Using the formula shown above, you can decode information contained in any sinusoidal function. The video below show how this is done.