Law of Sines and Cosines

We have come a long way this semester and have covered a great deal of information.  While this blog is aimed at helping us with reference and visualizing trig concepts in new ways, we must apply these principles to the real world.

In previous posts, wave-like functions were mentioned, and wave-like patterns in the natural world can be represented with trig functions.  Sine and cosine, especially, are concepts used heavily in physics for measuring force, work done (force times mass), and special applications for light and sound waves.  We have looked mainly at right triangles, but sine and cosine are not limited to this.  Here we introduce two more very significant laws.  The Law of Sines, and the Law of Cosines.

Let us first consider the Law of Sines:

Law of Sines
Retrieved from http://www.mathwarehouse.com/trigonometry/law-of-sines/images/formula-picture-law-of-sines2.png

The Law of Sines is a simple ratio, and with a little practice it should become free points on a test. Each angle is opposite its related side, i.e. angle A relates to side a.  We need to know any 3 of these elements to find the fourth.  

If we have side length a, angle A, and side b for example, we can find angle B.  HOWEVER, there is a big catch with this relationship.  Notice how a relates to sin(A), b relates to sin(B), and c relates to sin(C).  If we have any 3 elements, we might assume we can use the Law of Sines, but say we have length a, Length b, and angle C, we CAN NOT calculate any additional information, because we can only connect two letters like a/A to b/x or c/x.

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Now let us consider the Law of Cosines:

Using the same concept of sides a,b, and c, and angles A, B, and C, we comes up with a formula that is much more complicated, but is much easier to use in practice.  

Retrieved from http://mathworld.wolfram.com/LawofCosines.html

Notice this looks like 3 formulas, but it is really only one.  What is important to notice is, whatever side we are solving for, we add the OTHER two, then subtract the quantity (2 times the other two, times the cosine of the original).  That is too complicated in words, but:

We START with "a" and END with "A".  The rest of the formula is b's and c's.  

It looks closely related to the Pythagorean Theorum, and that is with good reason.  If you are curious about how this formula was derived, visit http://mathworld.wolfram.com/LawofCosines.html.  However, that won't be necessary for our course.  This formula is brilliant for solving the puzzle of missing sides and angles, and with a basic calculator and a sheet of scratch paper we can discover a great deal of information that is otherwise impossible to find - with absolute precision I might add.

Here is a great video explaining the details if you have the time.  Feel free to use it as a resource for your homework, or do a simple Youtube search for the laws:

"When to Use Law of Sines and Cosines"

This has infinite applications, but some historic examples range from finding the distance across a river or calculating the massive distance between planets.

Planets Example
Retrieved from https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbzMvnyMeaHB5CMImDlh8K86rGhkPR9aCv8VMsU94lfrIHgnsukJQEGzrs-8q8ZSIv2ncCm5bLyVhyphenhyphenXduT8RWIl-9ExE5dTT2j8dh32_SYBn85LqRNzr-_m_yapzgf-EEsezOIIaU7o5I/s320/triang1.gif

Island Example
Retrieved from https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3CsFZK2ec6WVgT4NPfhE3mGdsJ0EJJHDnYRGTslNaTQ4znEb5D102DgOAi1arT9L7g_Uopza7afIY6eBLvAERvJ6hH264_rZjpGci07Q6V5sqSieBcas2XmxQLnFFHP1c9RyU-agoSNg/s320/distancetoisland.jpg

The sky is the limit is certainly an understatement.  These laws are deceptively powerful tools for discovery, and are very simple to work with when we have worked through a couple examples!

Sources: 

Law of Cosines. (n.d.). Retrieved December 6, 2015, from http://mathworld.wolfram.com/LawofCosines.html

Why Learn Trigonometry?

We have spoken a great deal about visualizing functions and trigonometry concepts, but let's take a step back for a moment and see the bigger picture.

Trigonometry did not click for a long time for me.  To this day, I am still relearning the simplest ideas and seeing them in new ways.  The unit circle is nice to learn, as it gives us a great foundation and the number 1 is easy to work with, but there is another fascinating parallel we can make.

The height of the human body is roughly equal to our arm span.  If we draw a square and a circle around a human figure, we can see two very distinct relationships, and this is where ancient mathematicians discovered a great deal of their insight and wisdom.

Vitruvian Man
Retrieved from https://en.wikipedia.org/wiki/Vitruvian_Man#/media/File:Da_Vinci_Vitruve_Luc_Viatour.jpg

If we stand with legs together and arms straight out, we form a square, and if we angle our legs and arms, we maintain the same relationship, but since we are moving a constant radius, we are moving our limbs along a circle.

If you have trouble memorizing the trig functions in the way we discussed earlier, perhaps we can visualize them another way.

Pretend we are in the middle of a dome.

Now pretend someone has built a wall at the edge of our dome.  It would appear like this:

Wall Example
Retrived from http://betterexplained.com/wp-content/uploads/trig/trig-wall.png

In this example, we can visualize tangent and secant in a new way.  Suppose the wall is blocking our vision.  To climb the tangent wall (green), we'll build a secant ladder (red).  Now you can SEE, CAN'T you?  Get it?  Secant?  Tough crowd...

I didn't come up with it, but thank Kalid Azad at http://betterexplained.com/articles/intuitive-trigonometry/.  I highly recommend reading this site for a more thorough explanation.

Now that we have seen tangent and secant, cosecant and cotangent are not far off.  Instead of a wall, let's pretend a ceiling was built instead.  Imagine that it extends outward beyond the edge of the dome.

Ceiling Example
Retrieved from http://betterexplained.com/wp-content/uploads/trig/trig-ceiling.png

In this example, the height of the ceiling is the radius of the dome (green).  This could easily have been built at the origin, but placing it outward better illustrates the red ramp, which is the cosecant.  If we have a given angle, the cotangent is the length of the base of the ramp, and the cosecant is the length of the ramp itself, rising toward the ceiling.

When we put both of these drawings together, we get a brilliant illustration of all the trig functions relating to one another, all forming similar triangles which can easily be solved with simple ratios.

Combined Graph
Retrieved from http://betterexplained.com/wp-content/uploads/trig/trig-overall.png

From here, we can actually re-derive the trig identities we learned last time in a much simpler way than memorizing.  Take the blue segment, the green segment, and the red segment connecting the two, and we form all three of the Pythagorean Identities.

Consider the original Pythagorean Theorum:
a^2 + b^2 = c^2

cos^2(x) + sin^2(x) = 1

1 + tan^2(x) = sec^2(x)

cot^2(x) + 1 = csc^2(x)

It's pretty amazing when we put them all together in this way.  Trig functions can be derived in so many ways because triangles and circles are so closely related.

Sources: 

Kalid, Azad. How To Learn Trigonometry Intuitively. Retrieved November 30, 2015, from http://betterexplained.com/articles/intuitive-trigonometry

Graphing Trig Functions

Now that we have been introduced to the reciprocal trig functions, secant, cosecant, and cotangent, let's consider their graphs.

Recall with sine and cosine, we saw their relationship was very closely related, especially over the period of 2π when studying the unit circle.
I always remembered sine as a sideways "S", since it starts with S.

Similarly, for cosine, I always thought of it as a "Crow" or a bird, since it looks like a bird flying in the distance and it conveniently starts with a "C".

In any case, sine and cosine are simple functions to draw.  If we remember our unit circle values, sine is 0 at 0, π, and 2π, and is 1 at π/2 and 3π/2.  Draw a dot for each of those points like you would for any other function, and just do your best to connect the dots with a curve.  The cosine graph is the same, but where sine is 1, cos is 0, and where sine is 0, cos is 1.



What about secant and cosecant?  If we never knew what sine and cosine looked like, we would never be able to make sense of them, but let's see what they look like when considering their reciprocals.

Though they look like U's, they are actually distinct parabolas heading toward infinity.

Cosine = Adjacent / Hypotenuse.   When the Adjacent side is 0 on the unit circle, the angle is 90 degrees, and the hypotenuse is 1.  We can divide 0 by 1, and it equals 0.  What happens when you flip them?  We get 1 divided by 0, which is impossible.  Let's get as close to 0 as we can within reason though, and divide 1 by .000000001 - a very very small number.  We get 1000000000, a very very large number.  The closer and closer we get to 0, the larger and larger the number gets.  So we get a vertical asymptote everywhere that cosine = 0.

It is for this reason that we can simply draw parabolas on each bend of the sine and cosine curves.  I apologize for carrying on so long, the important thing to take from this is to understand why sine and cosecant are related, and cosine and secant are related.

To draw the curves (Let's graph sec(x)): 

1.)  Draw the reciprocal function (red).  Secant relates to cosine, so pencil in the graph of cosine along 0 to 2π.  We choose 0 and 2π because this is the period of the graph.

2.)  Pencil in the vertical asymptotes.  Wherever the graph of cosine is 0 on the y-axis, the secant will approach infinity.  These are the black lines in our sample graph.

3.)  Draw parabolas reflecting out of the graphs curves (shown in green).  These are the periodic minimums and maximums of the graph, also known as the magnitude.  The magnitude for trig functions on the unit circle are simply 1.

The accuracy of hand-drawn graphs is not particularly important, as long as you understand the values where the graph hits 0 and 1, that is what is important - most notably at 0, π/2, π, 3π/2, and 2π.


Graphs created with Desmos Graphic Calculator.
https://www.desmos.com/calculator

Secant, Cosecant, and Cotangent : Simplifying Trig Functions

As we explore trig identities and their graphs further, it is important to emphasize how each of them relate to one another.  When we look at any given right triangle, we know that sine = the opposite side (from a chosen angle) over the hypotenuse, which is the longest side.  Cosine = adjacent over hypotenuse, and tangent = opposite over adjacent.

SOH CAH TOA retrived from:
http://mathworld.wolfram.com/SOHCAHTOA.html

SOH - CAH - TOA is an extremely useful mnemonic to remember these relationships, and it is likely you have already been introduced to this simple reminder.  However, if we flip those fractions, we create three more unique functions.

Sine (sin) -->  Cosecant (csc)  :  sin=O/H  --> csc=H/O

Cosine (cos) --> Secant (sec)  :  cos=A/H --> sec=H/A

Tangent (tan)  -->  Cotangent (cot)  :  tan=O/A --> cot=A/O

It always helped me to remember if we flip the fraction, you add "co" to the beginning, or take it away.  Sine becomes CO-secant.  Cosine already has "co", so we take it away, and it becomes secant. Tangent turns to CO-tangent.

These are derived by using laws of multiplying and dividing fractions.  Using trig identities, we can easily cancel functions out and simply many hideous and scary looking formulas.  By writing all of these functions in terms of sine and cosine, we can work with something much easier and more familiar to us.  We'll prove them in class, but for quick reference:

tan = sin/cos, thus

cot = cos/sin

Also, remember that as these are reciprocals, remember that these functions can be written as 1 over their reciprocal.  

For example:  
1 / sin = csc
1 / cos = sec
1 / tan = cot
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3 of the most important trig identities are based off these laws, and thankfully there are more dorky ways to remember them.  These are the Pythagorean Identities, and are derived from playing around with the simple Pythagorean Theorum we all know;  a^2 + b^2 = c^2

Retrieved from
http://www.rachel.worldpossible.org/modules/olpc/wikislice-en/files/articles/Trigonometry.htm

sin^2 + cos^2 = 1, we simply have to remember, and it will become the most familiar identity.  The other two are not as simple though.  Here is how I always remembered them:

1+tan = sec -->  1 TAN man has a SECret.

1+cot = csc -->  1 man on a COT Cannot Sleep Comfortably.

It's a bit of a stretch, and the first time I heard it I hated it, but it was so ridiculous that I remembered it for the rest of my life.  All that is important is the 1+TAN=SEC, and 1+COT=CSC, and to remember that these quantities are squared.  If this doesn't work for you, feel free to memorize it any way that works for you.
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Preview and Practice: (http://home.windstream.net/okrebs/page84.html)

This site has a few brilliant examples of how we can rewrite in terms of sine and cosine, to begin to get a handle on how these functions relate to one another.  It's hard to see why this skill is useful right now, but trig identities become a very powerful tool in higher mathematics - especially if you plan on continuing to study through calculus.

Sine and Cosine: How do they interact?

Hello again, class!  I'd like for us to take a closer look at the graphs of sine and cosine functions. What can we learn from studying their relationship? We know that they form a circle somehow; and as one increases one decreases, but why is that? Hopefully we make life a little easier for you in the coming lessons by visualizing them.

When we study the unit circle, we see how sine and cosine are directly related in terms of x-y coordinates.  This is the rectangular coordinate system that we are used to, where we draw a grid.

Unit Circle
http://mathmistakes.info/facts/TrigFacts/learn/images/ucdefp.gif

When we start at 0, the value of cosine is 1, and sine is 0.  As we increase the angle, we climb the circle upwards, or increase the sine.  We notice from here that the cosine begins to shrink until we hit 90 degrees, where sine is now 1, and cosine is now 0.  As we continue along the circle, sine shrinks and cosine starts going into negative values until we hit 180 degrees.  Continuing along, sine and cosine are both negative and finally, we enter the 4th quadrant where the circle closes up.

Sine and Cosine (+ and -)
http://aventalearning.com/courses/ALG2x-HS-A09/a/unit05/resources/images/A2_5_3_10_unitCircle.jpg

This relationship might be hard to conceptualize at first, but here is a brilliant image of a machine that roughly illustrates sine and cosine's relationship in the Unit Circle.

Sine and Cosine.
  http://www.businessinsider.com/7-gifs-trigonometry-sine-cosine-2013-5

Notice how this relationship continues on infinitely.  The domain, or set of inputs that defines a function, is defined for all values of theta.  That means we could calculate the sine and cosine values for -360000000 degrees, and 360000000 degrees, or any infinitely large value you could imagine.

Practice: Try putting a random number into your calculator and pressing the sin key.  For example, let's try 60.

sin(60)=0.8660254...

Let's add 360 degrees to that, or a complete turn around the circle.  360+60=420

sin(420)=0.8660254...

Let's add 360x100 to that.  360x100 = 36000+420 = 36420.

sin(36420)=0.8660254...

We get the same result!  You calculator is actually solving for unit circle values for these functions. Consider the following image as well:

Sine and Cosine
http://static4.businessinsider.com/image/51910f38eab8ea4c31000002/0tlnknd%20-%20imgur.gif

We can see, as the circle is traced, it's "shadow" or "projection" on the yz axis (drawn in red) is actually the sine function, and its projection on the xz axis (drawn in blue) is the cosine function! When we visualize it this way, we see how these infinitely occuring functions occur - it is because we can trace infinitely many circles in space.

Doesn't it look like a spring?  It's not a far stretch to assume that trig functions can be applied to physics and mechanics.  That is only the beginning - the usefulness of these functions goes on forever, just like the functions themselves.

Sound acts in waves - light acts in waves; there is so much about the world we can discover through these ideas!  Any circular pattern we observe in the world can likely be modeled and adapted to trigonometry; even those in nature.  Concepts like shadows, hours of sunlight, and even wildlife population can be studied with trigonometry.

Learning the Unit Circle

Greetings everyone, Mr. Keen here.

As we dive into Trigonometry this year, we are going to be met with a ton of new concepts.  The unit circle is one such concept which will become extremely useful in our class.  It is so useful, in fact, that it will continue to be used throughout calculus and beyond.

Some of these concepts seem strange to work with at first; maybe even intimidating.  Instead of polygons, we start seeing curved functions like sine, cosine and tangent, which suddenly seem a lot more difficult to work with.  The circle is an easy and familiar shape though.

We have a good handle on working with rectangles and right triangles. The unit circle is really no different: we are just looking at right triangles from a new perspective. The definition of a circle is all points equidistant from its center, with the distance being r (the radius). We know 2πr is the circumference of any circle, and the unit circle is simply a circle with r=1, giving us a circumference of simply 2π.

When we draw the unit circle at the origin (0,0) it is divided into four 90-degree quadrants.

Unit Circle: http://etc.usf.edu/

The beauty of the unit circle is you only need to memorize the first quadrant.  We have 5 angles to work with, which are 0°, 30°, 45°, 60°, and 90°.  We can trace a right triangle over all of these.

Lets look at 0°.  On a coordinate plane, we simply get the point (1,0) - this is because the radius (1) is going straight in the x direction.  Easy, right?  We can always think of cosine as the "x" value and sine as the "y" value in rectangular coordinates.  When I was in school I was taught to memorize all of these values, but simply knowing this x - y relationship is more than half the battle.  If you understand this, you're better off than I was in college.

Remember in Geometry how we found sine and cosine with SOH-CAH-TOA? Sine=Opposite/Hypotenuse and Cosine=Adjacent/Hypotenuse.  Well in the case of the unit circle the hypotenuse (which is always the radius) is always 1, therefore we can ignore it.  So sin=opposite, cos=adjacent, or sin=y, cos=x.

Circle Diagram: Unit Circle & Evaluate Trig Functions - Mr. Gais

Let's look at 30°.  Notice it is further in the x direction than the y direction.  We see (√3/2, 1/2). Notice √3/2 is a larger quantity than 1/2, and in the y direction it appears to be halfway up the circle, as we would expect.

Let's look at 45°.  This forms a 45°-45°-90° isosceles triangle, so the x and y values are the same!    It happens to be (√2/2, √2/2), which is the only other number you need to memorize!

Finally let's look at 60°.  This angle is very similar to 30°, but the x and y values are switched, giving us (1/2, √3/2).

The angle 90° is much like 0°.  There is no x value, and the y value is simply 1, because it is the radius going straight up in the y direction touching the very top of the circle, giving us the point (0,1).

Now that we have these values, we can easily illustrate the rest of the unit circle by placing negative signs in the appropriate places.  In the 2nd quadrant, the x value is negative, but the y value stays positive.  In the 3rd, both are negative; in the 4th, x is positive and y is negative.

Quadrants: MathIsFun.com

Q1: (+,+)

Q2: (-,+)

Q3: (-,-)

Q4: (+,-)

That's all there is to it!  We will practice soon with reference angles, but let's do a quick example ahead of time.

Find the unit circle values for 225°.  Let's think for a moment.

225° ends with a 5.  Surely our best guess would be that it is a reference angle of 45°, since 225° isn't divisible by 30° or 60°.  If we visualize it, we rotate counterclockwise starting at the angle 0.

Image from http://math.stackexchange.com

180° is halfway around the circle.  225°-180° = 45°.  If we go 45° further along the circle, we arrive in the 3rd quadrant with a 45° reference angle.  The unit circle value for a 45° angle is (√2/2, √2/2).  Since we are in the 3rd quadrant, both values are negative, thus our answer is:

(-√2/2, -√2/2)

As long as you know how far from 0 or 180 the angle is, you can easily find its unit circle values.

For those who like to use their hands to help visualize, there is an amazing trick that makes learning the unit circle as easy as counting 1,2,3; literally.  YES you can use this during a test! Here's how:

Left Hand Trick - https://www.youtube.com/watch?v=LE6dmczMc68

Sources:

Gais, J. (n.d.). 2.0 - Unit Circle & Evaluate Trig Functions - Mr. Gais. Retrieved September 28, 2015, from https://sites.google.com/site/mrjgais/Home/trigonometry/unit-circle 

Pruitt-Britton, M. (n.d.). Memorizing the Unit Circle Using Left Hand Trick. Retrieved September 28, 2015, from https://www.youtube.com/watch?v=LE6dmczMc68 

Rajpoot, H. (n.d.). Calculating a Point that lies on an Ellipse given an Angle. Retrieved September 28, 2015, from http://math.stackexchange.com/questions/22064/calculating-a-point-that-lies-on-an-ellipse-given-an-angle

Trig Unit Circle Why? (n.d.). Retrieved September 28, 2015, from https://www.physicsforums.com/threads/trig-unit-circle-why.475575/ 

Unit Circle. (n.d.). Retrieved September 28, 2015, from 

http://etc.usf.edu/clipart/43200/43215/unit-circle7_43215_md.gif