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You Are Impossible Without Trig

Trigonometric Functions are Essential to Everything You Do

You Are ImpossibleWithout TrigTrigonometric Functions are Essential to Everything You Do

Right Triangle Trigonometry

The six trig functions above can be used to find unknown values for a right triangle. You only need to know Three things about a triangle to find everything else using trig. These three things are:
  • One Side Measure
  • Another Side Measure
  • That The Triangle is Right

For example, sin applied to this triangle means that the measure of the opposite side is equal to the measure of the hypotenuse multiplied by the sin of angle A. To better illustrate this, and more, you can view the following images:

Solving For a Side

Solving For An Angle

When solving for angles, inverse functions may be used. These are denoted sin^-1, or which ever function you are inversing. The use of inverse functions can be seen above.


Finally, the symbol theta is a universal symbol to represent angle measure. You will see it later in the newsletter, and looks like a 0 with a horizontal line through the middle of it.

The Unit Circle

The Great Unit Circle Extravaganza!

Graphing

To graph, as mentioned in the Unit Circle Section, take the points around the Unit Circle, plot them, and draw a smooth curve between them.

f(x) = sin(x)

Notice how the starting point, that is, the point at x=0, is at the origin. In a sinusoidal (which includes both f(x) = sin(x) and f(x) = cos(x)) graph, the only distinguishing feature is the starting point on a function which has not undergone a phase change. In sine graphs, the starting point will be at (0,0) for functions which have not horizontally shifted.

Also notice how the first peak in quadrant one is at y=1. This corresponds with the point on the Unit Circle which matches to the angle pi/2, which is the x value. Keep in mind that although the x-axis is labeled in integers, the function is still graphed in radians. The radian points are just simplified fully, meaning the numbers were multiplied by the value of pi. This can be seen on the graph when the function crosses the x axis at about 3.14, or pi radians.

f(x) = cos(x)

Notice how the starting point for cosine functions which have not undergone a phase shift is at y = 1.

How To Transform A Sinusoidal Function

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Amplitude = 1

f(x) = sin(x)

Parent Function

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Amplitude = 2

f(x) = 2sin(x)

Vertically Stretched

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Amplitude = .5

f(x) = .5sin(x)

Vertically Compressed

When a =/= 1, add the amplitude to the y value of the midline to get the y value for the maximum. In addition, you should subtract the value of the amplitude from the y value of the midline to get the minimum. Amplitude only changes maxes and mins, and does nothing to the midline.
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Frequency = 1

f(x) = sin(x)

Parent Function

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Frequency = 2

f(x) = sin(2x)

Horizontally Compressed

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Frequency = .5

f(x) = sin(.5x)

Horizontally Stretched

When b =/= 1, use the period formula. A period is the part of the graph in between two corresponding points, like two peaks of the wave. Another way of defining period is the longest stretch of the graph possible before you start repeating patterns. The formula for period is 2pi/b. To find the new period when b =/= 1, plug b in. This is your new period, and you can travel this value far away from the starting point along the x axis to get a point on the graph. From there, to get more points, you can find 1/4, 1/2, and 3/4 of the value of a period to find the x-value of other points within the period.
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Phase Shift = 0

f(x) = sin(x)

Parent Function

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Phase Shift = -1

f(x) = sin(x-1)

Phase Shift to RIGHT

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Phase Shift = +1

f(x) = sin(x+1)

Phase Shift to LEFT

When c =/= 0, follow these rules for phase shift. Phase shift values may be in integers, or radians. Whatever form, you simply add or subtract that value from the x values of every point on the function, depending on whether c was positive or negative. If c was positive in the function, you will subtract that value from all points, and the function will move to the LEFT. If c was negative in the function, you will add that value to all points, and the function will move to the RIGHT.
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Midline + 0

f(x) = sin(x)

Parent Function

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Midline + 1

f(x) = sin(x) + 1

Function Shifted Up 1

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Midline - 1

f(x) = sin(x) - 1

Function Shifted Down 1

When d =/= 0, the equation of the midline is equal to whatever the new value of d is. In addition, every point's y value will increase or decrease by the value of d depending on whether it is positive or negative, respectively. In the parent function, the midline is at y = 0. The midline is ALWAYS equidistant from the maximum and the minimum.

The Long Forgotten Tangent Function

A tangent function is a function with amyptotes. In the form of f(x) = a tan(bx), It has the following characteristics:


  • Domain: {x|x =/= 90 + 180n, where n is an integer}
  • Range: {All Real Numbers}
  • Amplitude: Undefined
  • Period: 180 Degrees


The asymptotes of the parent tangent function are multiples of 180, but do not lie on 180, 360, 540, etc., but instead lie on 90, 270, 450, etc.


Tangent Functions are transformed in much the same way as sinusoidal functions. Increasing frequency horizontally compresses them, thereby getting more repeated lines per given interval. It can be shifted horizontally and vertically the same a sinusoidal graphs. The only difference is that tangent does not have an amplitude, since it has no maximums or minimums.

Here's Where the Impossibilities Come In

Electromagnetic Radiation and 3-D Glasses

Light, and every other type of electromagnetic radiation, travels in a wave. It follows a periodic, sinusoidal path. Increasing the frequency makes the wavelength shorter, and bluer, when talking about visible light (400 < Wavelength of Visible Light < 700, roughly). Decreasing the frequency makes the wavelength longer, and redder. Without trigonometry and the sinusoidal path that photons take, there would be no vision, no radio/cell phones, and no microwave ovens.
An interesting application of the sinusoidal nature of light is 3D glasses, or more specifically, anaglyph glasses, as shown above. These use a red filter for the left eye, which permits only red wavelengths to pass through, and blocks blue and green. They also use a cyan (the combination of blue and green) for the right eye, to allow blue and green to pass through, but not red. By using anaglyph images, which effectively double the image, offset them, and apply either a red or cyan filter to each, each eye will only see its corresponding image. Thus, different frequencies of light allow for anaglyph steriopsis.
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Stationary String

The string is stationary, and there are no pressure waves traveling. Consistent Pressure.
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Compression

The string moves forward, and pushes air particles into the particles farther to the right.
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Rarefaction

The string moves backwards, and leaves a space of low air pressure where it was in the last step. Air rushes back into this space, but only after pushing on other air particles. This occurs over and over, moving further to the right. Thus, a wave

Sound and Noise Cancellation

Unlike light, sound is not a form of electromagnetic radiation. It is merely just vibrations in the air. It does, however, travel in a wave - a sinusoidal pattern. As such, hearing would be impossible without trig functions.


Changing the frequency of sound affects the pitch of the sound. Amplitude affects the volume. Then what does phase shift do? If you take two identical trig functions, and line their peaks up perfectly, the volume will be doubled. Interestingly, if you take those same two identical trig functions, and shift one over by pi, you will perfectly line up the peak of one function with the trough of another. This completely eliminates sound, setting amplitude equal to zero, and is the premise behind noise cancelling headphones.

Topics In Summary

Right Triangle Trig

By using the 6 trigonometric functions, you can find missing values of right triangles by only having two sides and one angle (the 90 degree one!). Basic equations skills along with the use of inverse functions can yield missing side lengths or missing angle measures.

Unit Circle

The Unit Circle is the combination of the trigonometric functions onto a circle, with distinct points. This circle draws the connection between algebraic trigonometry, and graphical trigonometry by corresponding angle measures with function solutions.

Graphing

Sinusoidal functions form a wave while Tangent functions have anymptotes. They can be transformed by altering their amplitudes (only on sinusoidal functions though), frequencies, phases, and vertical shifts. The x axis is represented by angle measures for these functions, whether that be in Radians or Degrees.

Upcoming Tests & Quizzes

Quiz 4: Dec. 7
Test 2: Dec. 10

Topics: Exponential Functions, Logarithms, and Conics (Woo!)

Citations