PhysicsGutka

Reciprocal identities Pythagorean Identities Quotient Identities Co-Function Identities Even-Odd Identities Sum-Difference Formulas Double Angle Formulas Power-Reducing/Half Angle Formulas Sum-to-Product Formulas Product-to-Sum Formulas   **ART  If you do remember all of them good. But if you face any problems recalling them we have a trick.

Binomial Expansion For any value of n, whether positive, negative, integer or non-integer, the value of the nth power of a binomial is given by:      In many derivation we are using Binomial expansion to simplify the expression or formula.   Important Expansion         Summation   Sum of first N numbers = n(n+1)/2 Sum of squares of N numbers = n(n+1)(2n+1)/6

Graph for   Graph for   Graph for

Graph for y = cos x We all are aware about how the values of sine changes when angles are changes. Thus we can say Values of cos x lies between -1 to +1   Graph for y = sine x Values of sin x lies between -1 to +1 i.e. y will vary from -1 to +1    These Graphs are important as per the Point of view of SHM and Wave theory.  

Example 8   Graph   and . Solution There really isn’t a lot to this problem other than making sure that both of these exponentials are graphed somewhere.  These will both show up with some regularity in later sections and their behavior as x goes to both plus and minus infinity will be needed and from this graph we can clearly see this behavior. Same graph you may have observed in Radioactivity where a Radioactive Element decays as it is represented by blue line. Again on multiplication with negative graph will be reversed as the were done in previous examples.

Example 7   Graph Solution This is a hyperbola.  There are actually two standard forms for a hyperbola.  Here are the basics for each form. Form Center ( h, k ) ( h, k ) Opens Opens right and left Opens up and down Vertices a units right and left of center. b units up and down from center. Slope of Asymptotes So, what does all this mean?  First, notice that one of the terms is positive and the other is negative.  This will determine which direction the two parts of the hyperbola open.  If the x term is positive the hyperbola opens left and right.  Likewise, if the y term is positive the parabola opens up and down. Both have the same “center”.  Note that hyperbolas don’...

Example 6   Graph Solution The standard form of the ellipse is                                                           This is an ellipse with center ( h, k ) and the right most and left most points are a distance of a away from the center and the top most and bottom most points are a distance of b away from the center. The ellipse for this problem has center (2, -2) and has  and .  Note that to get the b we’re really rewriting the equation as,     to get it into standard from.  Here’s a sketch of the ellipse.        **ART : Standard equation of ellipse given above shows ( a is semi major axis and b is sem...