Advanced Math/Polynomials and trigonometry BIG problem
In this project you will investigate the feasibility of approximating the sine and cosine functions using polynomials. There is a general method using calculus to determine how closely a function can be approximated by using polynomials. You will investigate how well polynomials you drive approximate the sine and cosine functions.
You are lol to follow the steps below. You do not need to do the steps in exactly the order indicated as long as you cover everything you are asked to do after completing the outline below you should write your results and calculations neatly be sure to include relevant Grafton what you turn in. Your grade will be determined in part by your presentation.
Elmer is writing and educational computer program for kids. The program contains bright graphicsand cute sounds. In order to make sectors of circles appear on the screen the program needs to be able to compute sin x and cos x for random values of x. Elmer is using C++ to write tge program. In his version of C++, if he includes the standard library of math functions it will increase his code size beyond what would be feasible for the market. He therefore wishes to write his own functions in C++ which would give him a good approximation for sin x and cos x, so he does not need to include the whole math library. He wishes to find polynomials p(x) and q(x) so that |p(x)-sin(x)|< .0005 and |q(x)-cos x| < .0005 for any value of x between -π/2 and π/2. Elmer thinks he remembers from his calculus course several years ago that for any desired degree of accuracy is possible to find two polynomials to approximate the sin and cos functions for every real value x. his daughter Anna, who is taking pre calculus says that would be impossible. Who is right? Is it possible for Elmer define polynomials p and qwith the properties given above? If so what are the lowest degree polynomials he can find that meet the condition ?
1. Plot sin x and mx+b on the same axesfor various values of m and B. Experiment until you find the values for m and b which give a line that seems to be the best approximation for sin x when x is very close to 0. Do the same for cos x.
2. the next step is to prove the approximation for sin( theta) found in part 1 when theta is close to 0 is a good approximation. Follow the steps below
A. draw a sector of a circle of radius one having angel theta with 1 ray on the x axis and the other in the first quadrant. draw 2 right triangles using the angle of the sector as one angle. what the first triangle put the right angle at the point (1,0). For the second, drop the perpendicular from the point on the circle meeting the other side of the sector to the x axis. B. Compare the area of the sector with the areas of the triangles to get inequalities. C. Simplify the inequalities found in b to get the approximation for sin theta. the Apes what this proves your approximate Shin in part 1 is a good approximation.
3. next you will use what you learned part 1 to find polynomial approximations for sin x and cos x. the philosophy you should follow is that you have a linear approximation for sin x and cos x when x is close to zero. Use the double angle formula to rewrite sin x in terms of trig functions of angles smaller than x. at this point you can approximate all the occurrences of sin and cos with the linear approximations you found above. Try using the double angle formula several times before making the approximation to see if you get a better polynomial approximation. Do the same for cos. ( mathematica will simplify your calculations.)
4. Plot Some graphs to see how close to approximations you found part 2 are to sin x and cos x.
5. try removing some of the highest order terms to see if you lose much accuracy in the approximation. What are the lowest degree polynomials you can find that meat Elmers conditions for approximating the sine and cos functions? Plot the graphs to show that the error is acceptable to Elmer.
6. Try to approximate pi using what you found above.
7. Who is right Elmer or Anna ? discuss why.
8. Discuss why you would expect the process you did in step 3 to work.
Please, please, help me solve this its due monday and I dont even know where to begin.
Whoa, usually we don't answer homework problems but I can at least get you started. A complete answer is also hard because parts of the question are garbled and I don't really know exactly what you are expected to know (Mathematica?). Nonetheless...
For Part 1, near x = 0, the straight lines you should come up with are y = x for sin (i.e., m = 1 and b = 0) and y = 1 for cos. I'm not sure what your teacher had in mind in terms of you experimenting with m and b but these solutions can be seen by inspection (i.e., just looking at the graphs of sin and cos).
Part 2 is a little garbled (the radius of the circle is 1?) but you should be able to show that the areas of the little and big triangles approach the area of the sector, = theta, as theta goes to zero. Try using sin(theta) ~ theta= rise/hypotenuse.
Part 3 uses the double angle formula cos(x) = 1 - 2(sin(x/2))^2, which if you use the approximation in Part 1 of sin(x/2) ~ x/2 becomes
cos(x) ~ 1 - 2(x/2)^2 = 1 - (1/2)x^2.
Similar manipulations can bootstrap you to higher powers of x for sin and cos.
You might want to look up the Taylor series expansions (polynomials in increasing powers of x) of sin and cos to help guide your calculations (I assume this is legal for your course).
The rest of the work is mostly arithemetic which I'm sure you can do.
Please try your best with the above hints and send me a follow-up showing your work if you need more help.