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The Grandaddy of Trig Identities!

Sine Cosine Tangent

The challenged was issued by the most brilliant of educators: BlueCereal – a genius among us smart folks!  He suggested a post about our favorite content – challenge accepted Mr. Cereal!

I taught high school math for over 18 years before embarking on my current journey as a full time Ph.D student at Oklahoma State University.  During that time I taught almost every math course offered in high school: Geometry, Applied Geometry, Applied Math II and III, Advanced Algebra Trig, PreAP Precalculus, Calculus, Math Concepts, Algebra II, PreAP Algebra II, Algebra I, PreAP Algebra I, Algebra I-First Half, Algebra II Support…You get the picture – lots of math.

Across all of those classes there is one I just love to teach – Calculus!  I could teach this course 24/7/365!  It is so darn interesting!  The relationship of Calculus all the back to the beginnings of Algebra I is fascinating!  I could go on and on…

However, my most favorite lesson to teach, and its not even close, is the pythagorean trigonometric identities.  These things have fascinated me since Lu Ireton introduced them to me in her Math Analysis class my junior year of high school.  She is the teacher who propelled me to where I am today!  THANK YOU Mrs. Ireton!!

Pythagorean TheoremYou ready for a quick lesson?  Let us first revisit the pythagorean theorem from Geometry.  When you have a right triangle (angle triangle with a 90 degree angle), then you can use the lengths of the legs to determine the length of the hypotenuse.  We will use this basic Geometry to help us with the more complicated trigonometric parts.
Cartesian Right TriangleSince a, b, and c are all variables, we can use any letter (or thing) to represent them.  For our trigonometric purposes we will use the horizontal leg as x, the vertical leg as y, and the hypotenuse as r.  This image shows how we could represent this triangle on the coordinate axis.  The point (x, y) represents the two legs of our right triangle and these two legs create the angle θ.  If you were to draw a perpendicular line from the point (x, y) to the x-axis you would have a right triangle.

Our equation has changed just a little bit to  Pythagorean Theorem   Hang on – this is where stuff just gets super duper amazing!  Let’s investigate the specific value of r = 1, this specific value will help some patterns be more visible.  When comparing the side of the triangle with xθ, and r the ratio created is the cosine – this shows what that looks like, remember Cosinewe chose r = 1.

Using the same steps again, this time comparing yθ, and r we get a different ratio called the sine, don’t forget that we are choosing r = 1.Sine

Notice in both of these instances what is happening, we are reducing the ratio (r = 1) and have an equality for both the sine and the cosine.

cos θ = x and sin θ = y

When you have equality, you can exchange the items that are equal.  Using x squared and replacing only those things that are equal (see cosine and sine equality statements above) – you have a new and very powerful trigonometric identity!

Pythagorean Trig Identity

This identity is the foundation for all other trigonometric identities!  This identity can be used to make complicated problems simple!

Did I make a typo?  Have a question?  Want to expound on how brilliant this identity is?  Leave a comment below…


About Scott

My name is Scott. After 18.5 years as a high school math teacher in public education I have made the move to become a full time PhD student. This decisions was difficult, but has been one of the most rewarding things that I have ever done. Teaching in high school was an incredible experience for me, so leaving an environment that I loved for the unknown was a challenge. As I high school teacher, I taught almost every math course that could be offered. I was able to earn National Board Certification in Young Adult Math. I was honored as my building Teacher of the Year, no mean feat at Edmond Memorial High School!! My career changed as I became fascinated with educational technology and all of the things that it can do for teachers. I flipped my class. I used iPads and blogging (in high school math!!). I started using gamification and mastery learning. I changed my practice. I chose to go back to school to learn as much as I could. To bring that knowledge from academia and research to the teacher on the front line. I have had the opportunity to present at several conferences and share what I have learned with others. Its through these connections that we can be the best teachers we can for our students. They deserve it and we sell ourselves short when we don't give it. I love talking with teachers about change. About incorporating educational technology. About the power that they have to change lives. My blog space is me, it shares my passions and frustrations, my joys and my learnings. If you are interested in what I am studying, please visit my graduate school pages. If you are interested in the flipped classroom, I have some links to get your started. I would love to meet you! Do not hesitate to reach out! I would enjoy the opportunity to work with your staff or trade ideas with your teachers - let me know! Have a great day! #BeBrilliant


2 thoughts on “The Grandaddy of Trig Identities!

  1. Scott,

    I too have been fascinated with the Pythagorean identity and its elegance for quite some time. I’ve always enjoyed inspecting {{\sin }^{2}}\theta +{{\cos }^{2}}\theta and looking at the result for any value of r. If we start with the definitions for sine and cosine as you have defined above, then we end up with {{\left( \frac{y}{r} \right)}^{2}}+{{\left( \frac{x}{r} \right)}^{2}}

    Once we simpify, we can see that \frac{{{x}^{2}}+{{y}^{2}}}{{{r}^{2}}}

    Finally, since we started with {{x}^{2}}+{{y}^{2}}={{r}^{2}}, we can just substitute in and end up with \frac{{{r}^{2}}}{{{r}^{2}}} which we know is 1.

    This same derivation can be done with using \sin \theta =\frac{opp}{hyp} and \cos \theta =\frac{adj}{hyp}.

    On a side note, I also really enjoy that there is no need to remember the other Pythagorean identities as they can just be derived by dividing both sides by {{\cos }^{2}}\theta or {{\sin }^{2}}\theta .
    Math is beautiful!


    Liked by 1 person

    Posted by Jeremiah Simmons | April 12, 2015, 4:01 pm

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