Introduction Welcome to my Developing Expert Voices project. In the following slides you will come across four questions. They all have to do with trigonometric identities so see if you can solve them. If you are unsuccessful, there are solutions there to help you out. Also for your convenience, there are two slides of trigonometric identities for you to use and look at when solving the problems. They are at the end of the presentation. Good luck and have fun. Here you are…
Solution One By looking at the numerator of the expression we want to change it into it’s inverse (giving each expression a home). Then we also change the denominator into their inverses but still keeping the addition sign. Also looking at the exponent, we notice the two identities that are equal to one. Next you multiply out the numerator to combine the two expressions. We can also find a common denominator in the denominator of the original expression (when finding a common denominator watch what you put as the numerator). The exponent can now become two. Cleaning up the entire expression, you multiply the numerator by the inverse of the denominator. Now that you have two fractions being multiplied together in the expression, you notice the identity and set the denominator of the second fraction to one. Multiply the fractions out to get one fraction. Reduce a sine and a cosine.
Solution One…Continued Look to the other side of the question and take the inverse of what ever is in the brackets (you want two fractions being multiplied out). Next you multiply it out to get one fraction. Noticing the exponent, you expand and have the expression as the fraction multiplied by it self once. Multiply and simplify. Expand the denominator noticing the two identities. Expanding once again using two more identities and you get the answer.
Solution Two Find a common denominator and multiply out the numerator accordingly. Use brackets to help distinguish the right side of the numerator so that you will not get a conceptual error. Multiply out the numerator, keeping the right side in brackets to avoid confusion. Also multiply the denominator together. Next multiply the negative sign through the brackets on the right side of the numerator. Reduce what you can in the numerator and add the rest of the terms together. Expand the numerator by multiplying out a two.
Solution Two…Continued Recognize the double angle identity for sine and the one for cosine. Simplify to get tangent. Plug in the double angle identity for tangent. Multiply. Expand the numerator in terms of sine and cosine. Look to the other side of the question to get a hint on which identities to plug into the denominator so that the L.H.S. equals the R.H.S..
Solution Three Seeing that you have the cosine of three theta, you can expand it into two theta plus one theta. Looking back on all of your identities, you recognize the sum identity for cosine. You must also keep in mind and watch that extra theta. Next you plug in the double angle identities for sine and cosine. Now for the cosine identity you know which one to choose because you look to the answer and see that it contains only expressions that deal with cosine so the identity that you use is the one that only contains cosine. Expand the brackets by multiplying. Watch the negative sign when multiplying! Once again you must recognize that the answer contains only secant in it so you plug in the Pythagorean identity for sine.
Solution Three…Continued Multiplying out the brackets would be the next step but watch the negative sign because it becomes very confusing but important (when multiplying, use the negative sign on the 2 so that it is –2). Add like terms together. Express cosine in terms of secant keeping it in it’s own set of brackets. Multiply it out so that the L.H.S. equals the R.H.S..
Solution Four Starting off in the left set of brackets you can use one of the Pythagorean Identities. Next moving into the next set of brackets, you change everything in terms of sine and cosine. The following step is to change cosecant in the left set of brackets into one over sine. Also in the same step you can simplify the right set of brackets. Next you simplify once again on the left bracket and find a common denominator on the right bracket. Be careful because what ever you multiply the bottom half of the fraction by, you must also multiply the top half of that same fraction by the same value. Here you expand tangent and multiply it by the inverse of the bottom half of the fraction. That’s for the left bracket. For the right bracket, you clean it up to have only one fraction in it.
Solution Four…Continued For this step you must reduce like terms on the left and use an identity on the right for cosine to expand. Once again using that same identity just like in the previous step, you expand the left side this time and multiply on the right. (Watch the negative sign!) Simply simplify. Multiply so that the L.H.S. equals the R.H.S..
List Of Trigonometric Identities The Fundamental Identities… The Sum and Difference Identities…
List Of Trigonometric Identities The Pythagorean Identities… The Double Angle Identities…
Reflection Why did you choose the concepts you did to create your problem set? I chose the concepts that I did because I really enjoyed the unit about trigonometric identities and wanted to see what it was like to take on the role of a teacher. I wanted to see what it was like to explain some type of math problem to another person. How do these problems provide an overview of your best mathematical understanding of what you have learned so far? These problems show my best mathematical understanding of what I learned because they show how complicated these types of problems can be. I tried to put in some of the hardest problems into the project that our class never covered in class.
Reflection Did you learn anything from this assignment? I learned that explaining something in detail doesn’t come that easy. Also because this project was so long and took forever to do, I learned to be patient and organize my time better. Was it educationally valuable to you? I like to think that this project was educationally valuable because now I have no excuse to get any problems on trigonometric identities wrong. Working on the project is like a review for the exam, all that time put into the unit should put me in good shape for that part of the exam.