What is theorem is the basis for the fundamental or first trigonometric identity?
In middle school, everyone learns of Pythagoras theorem which is the sum of the squares of the two sides of a right triangle adds up to be the square of the hypotenuse. If both sides of this pythagoras equation is divided by the square of the hypotenuse, then what you essentially get is that the sum of the squares of the sine and cosine of an angle in the right triangle adds up to be 1.
Why should finding the slope of a function and the area under it be the basic aspects of calculus?
Basic arithmetic started with addition and subtraction. Then, it advanced into multiplication (As successive addition of the same number) and division (successive subtraction of the same number). In calculus (this will be clearer with a graph or drawing), we use infinitesimally small quantities for multiplication (or essential area or integration) and for division (essentially the tangent of the curve at that point). Hence basic arithmetic operations of division and multiplication of infinitesimally small quantities in an asymptotic limit leads to the definition of the basic calculus operations - namely - differentiation and integration
You are given these two balls, each tied to a meter long string of insignificant weight, on your either hands. Ball A - a heavy metal ball with a size like that of a ping pong ball Ball B - a lighter but larger basket ball. You gently swing ball A so that it oscillates covering an angle of 5 degrees on either side of the vertical. You swing ball B with more strength so that it oscillates an angle of 20 degrees on either side of the vertical. Which ball will have a longer period of oscillation and why?
The laws of oscillation of a pendulum states that for small enough oscillations and for an insignificant length of the string, the time period of the oscillation is independent of (a) the mass, size, shape, density of the pendulum (ball at the end of the string) (b) the amplitude of the oscillation. The equation for time period T is (derived/ proved in detail) to be 2* pi * sqrt(l/g) where l is the length of the pendulum. Since both balls A and B are tied with a meter long string (same length), they both will have the same period of oscillation.