Help with these goniometric equations?
cos(2θ) + cos²θ = 0.5 (1 - 2sin²θ) + (1 - sin²θ) = 0.5 2 - 3sin²θ = 0.5 3sin²θ = 1.5 sin²θ = 0.5 sinθ = ±(√2)/2 θ = π/4, 3π/4, 5π/4, 7π/4 ::::: 2/(tanθ + cotθ) = 2/(sinθ/cosθ + cosθ/sinθ) = 2/((sin²θ + cos²θ)/(sinθcosθ)) = 2/(1/(sinθcosθ)) = 2sinθcosθ = sin(2θ)
Is it necessary to learn Maths (Calculus, Trigonometry etc) to study Computer Science?
I don't think you should choose a computer science program based on what you wish to avoid. A more practical approach would be think about the kinds of problems you want to solve, in areas both in and out of computer science, then focus your studies in areas that give you the tools to solve those problems. In any case, few programmers that I work with have an advanced math background. You will want to have enough familiarity with applied math to understand what formulas to use in creating algorithms, or at least know enough to intelligently research it when needed, but that is covered pretty well in most programs, and algebra covers a lot of it. Also, it's too soon to judge yourself as permanently bad at math. While you may have struggled with it up to this point, that can be due to many factors, from ineffective teachers to a lack of motivation, or many other reasons. You may find that once you have sufficiently rich project to sink your teeth in to that you are able to dig in and learn the math required far more easily than you expect. And last, which program you pick is not written in stone. As you get more deeply in to it, you will see more clearly what area you have an affinity for and interest in, and can look for a more suitable degree program armed with that understanding. Keep in mind that computer science is a field that is very rapidly changing, and choosing it is choosing a career of constant study. There may be periods where you do the same thing for a while, but at the pace of this industry, the need to learn something new never really ends.
What are the relations between [math]\cos{x}[/math], [math]\sin{x}[/math] and [math] e^x[/math]?
The relationship is of course Euler’s Formula:[math](e^x)^i = \cos x + i \sin x[/math]The [math]i[/math]th power isn’t initially intuitive, but this helps: The approximation for small real [math]y[/math], [math]|y|\ll 1,[/math] is[math]e^y \approx 1 + y[/math]If we believe the same approximation holds for small imaginary exponents, we get for small [math]y[/math],[math]e^{iy} \approx 1+iy[/math][math]1+iy[/math] is a sliver of an angle just above [math]1+0i.[/math] The approximation tells us [math]e^{iy}[/math] is approximately on the unit circle; in fact it turns out to be exactly on the unit circle:[math]|e^{iy}| =1[/math]The angle of [math]1+iy[/math] is [math]\arctan \frac y 1[/math][math]\approx y.[/math] Again subsequent analysis shows this approximation to the angle of [math]e^{iy}[/math] to be equality. So we have[math]\angle e^{iy} = y[/math]We’ve come to see (at least approximately) why for small [math]y[/math], [math]e^{iy}[/math] is on the unit circle at angle [math]y.[/math]When we multiply complex numbers we multiply their magnitudes and add their angles. So raising a complex number to a power means multiplying the angle by the power. When we raise [math]e^{iy}[/math] to the [math]\frac x y [/math] power we end up at angle [math]y \cdot \frac x y = x.[/math][math](e^{iy})^{\frac x y} = e^{ix}[/math] is on the unit circle at angle [math]x[/math], so we must have[math]e^{ix} = \cos x + i \sin x[/math]Thus the relation between [math]e^x, \cos x[/math] and [math]\sin x[/math] is[math](e^x)^i = \cos x + i \sin x[/math]
What is the difference between geometry and trigonometry?
Other than geometry being a lot broader, the main difference is that trigonometry is computational. Trigonometry was developed after geometry for the purposes of astronomy.Both depend on distances and angles, but trigonometry uses the measurement of angles while geometry deals with angles only in terms of equality of angles and sums of angles.There are three theorems which are central to both. One is that the sum of the internal angles of any triangle is equal to two right angles. Another is the Pythagorean theorem. The third concerns similar triangles. It says if the angles of one triangle ABC are equal to the angles of another triangle A'B'C' (with angle A = angle A', angle B = angle B', and angle C = angle C'), then the sides are proportional,[math]\displaystyle\frac{AB}{A'B'}=\frac{BC}{B'C'}=\frac{CA}{C'A'}[/math]These three theorems are the basis of the trigonometry of right triangles. They imply that if triangle ABC is a right triangle with a right angle at C, and sides a, b, and c opposite angles A, B, and C, respectively, then the ratios a/c, b/c, and a/b only depend on the angle A.That allows the definition of the trig functions for acute angles, namely,[math]\displaystyle\sin A=\frac ac, \cos A=\frac bc, \tan A=\frac ab[/math]Then, by measuring angle A and one side of a the triangle, the lengths of the other sides can be computed by means of trig tables. Also, by measuring two sides of a right triangle, using the same trig tables in reverse, the angles can be computed.There are other corresponding theorems, but they look different in geometry than in trigonometry. For example, in geometry, there are three congruence theorems: side-side-side, side-angle-side, and angle-side-angle. Those along with a little geometry and algebra, allow methods to solve oblique triangles (that is, triangles that aren't right triangles). The specific computations are encoded in the law of sines and the law of cosines. Note that in order to deal with obtuse triangles, the trig functions for obtuse angles are defined the way that they are.
What are some math skills required for an entry level programmer in any language?
The answer to this question depends on the kind of development work you plan on doing. Even an entry-level game developer will have to know enough math to work with physics engines, and if you are developing software for financial or scientific applications, you will have to know enough math and statistics to understand the material you are working with.As a junior web developer, I can only say something about web development, which is probably one of the programming fields that require the least math knowledge. I would say that high school level math skills should suffice for most web development. Basic algebra and goniometry should suffice. It’s also useful to have a grasp of matrices. I am not talking advanced matrix algebra, just a basic understanding of what a matrix is, how to represent two-dimensional data using arrays, and stuff like that.Overall, I would say that, at least for web development, a sense of logic, a solid understanding of software design patterns and good communication skills are a lot more important than pure math skills.
I am 16 and I want to be an excellent mathematician. What should I do?
I’m not exactly what you’d call an excellent mathematician, but I love maths and I know some things that might help you:Practice frequently. Don’t let maths work pile up. It is one of the subjects you can’t cram in two days before the exam. The things you learn need to settle before you can build more things on top of them.Learn to like maths. Maths is not all about getting the same answer as the one that’s at the end of the textbook. Try to visualize what you learn. Trigonometry, sin and cos functions, statistics, Fibonacci… examples for all of those can be found in nature. Just look around. ViHart, a mathemusician, could help you with that. I recommend:andby her.Khan Academy. Whenever I didn’t understand a concept in math class, I look up a Khan Academy video lesson for it on YT. Sal Khan and his team made math videos on (mostly) all topics covered in schools. He explains beautifully and makes everything seem so obvious. He also gives examples.MIT released a bunch of their recorded math lessons via OpenCourseWare. Just Google or YT them, you’ll find them easily, so that could help.YouTube channel Numberphile. Super fun videos even if you don’t understand everything, and they’re about maths.Good luck!