Pre calculus formulas11/25/2023 ![]() ![]() Just making little substitutions into the one formula that we started with to get the formulas for the other three expressions. We just started with the identity for cos(A-B) and then we made some clever substitutions to figure out cos(A+B), sin(A+B), sin(A-B). In each one of those identities, we did not use anything external. That is exactly the subtraction formula for sin. ![]() 0406īut now, we started with sin(A-B) and we derived sin A cos B - cos A sin B. 0394Īctually I should have said plus because look we have sin (-B) and sin (-x) is -sin(x). This is sin A, cos (-B), cos does not even function so that is cos B + cos A. Now again, we are going to use the odd and even properties. 0365Īccording to the sin formula that we just proved, it is the sin of the first one x the cos of the second one which is (-B) + cos of the first one x sin of second one which is (-B). The point of that is that we can then invoke the sin formula that we just proved, we got sin(something +something). We will write this as sin, instead of writing it as a subtraction, we will think of it as adding a negative. 0330įinally, sin(A-B) we are going to do the same trick that we did for cos(A+B). That is where the addition formula for sin comes in. Now we got the addition formula for sin, because we started with sin(A+B) and we reduced it down to sin A cos B + cos A sin B. Now, sin(pi/2-A) using the co function identity at the second co function identity is cos A x B. 0273īut now, cos(pi/2-A) again using the co function identity is just sin A, sin A cos B. So, this is cos of the first term, cos(pi/2-A), cos of the second term is B + sin of the first term x sin of the second term. 0255Ĭos, I am going to substitute n instead of A-B, I have (pi-2)-(A-B). I'm going to use my cos subtraction formula, this one that we started with. I am going to group those two terms together and then -B, because it was minus the quantity of A+B. That is by the first co function identity. The way we do that is I have sin(A+B), I'm going to use the first co function identity and write that as cos((pi/2-(A+B)). Somehow we are going to use those to derive the sin formulas from the sin formulas. Those say that cos(pi/2)-x is the same as sin(X), sin(pi/2)-x is equal to cos(x). Now, I'm going to have to bring in the co function identities, let me remind you what those are. I was able to do that much more quickly than we were able to prove the original formula for cos(A-B). 0156īut look, now I got cos(A+B) is equal to cos A cos B - sin A sin B, that is the formula for cos(A+B). Now sin A and sin -B, sine is odd so sin(-B) is -sin B. Sine is an odd function, sin(-x) is equal to -sin(x), I got cos A and cos(-B), but cos(-B) is the same as cos B. That means cos(-x) is the same as cos(x). I'm just going to invoke this formula above except whenever I see a B, I will change it to (-B). ![]() The point of that is now I can use my subtraction formula. I'm gong to write in addition, in terms of a subtraction. I'm going to start with cos(A+B) and I'm going to write that as cos((A-(-B)). I want to derive the other three formulas. The cos(A+B) is equal to cos A cos B + sin A sin B, we are allowed use that. Let us remember what that formula is because we are allowed to use it now. Hopefully, it would be easier than the original proof of the cos(A-B) formula. 0031Īnd now that we got that available to us, we are going to start with that formula and we are going to try to derive all the others. We really did prove the cos(A-B) from scratch. We are not getting trapped in any circular loops of logic. We did it without using the other addition and subtraction formulas. If you remember back in the previous set of examples, we proved the formula for cos(A-B). This time we are going to use the formula for cos(A-B) and the co function identities to derive the other three addition and subtraction formulas. Ok we are here to try more examples of the addition and subtraction formulas. Graphing Functions, Window Settings, & Table of Values Section 17: Appendix: Graphing Calculators Instantaneous Slope & Tangents (Derivatives) Section 15: Sequences, Series, & Induction Section 12: Complex Numbers and Polar Coordinates Using Matrices to Solve Systems of Linear Equations Section 9: Systems of Equations and Inequalities Word Problems and Applications of Trigonometry Modified Sine Waves: Asin(Bx+C)+D and Acos(Bx+C)+DĬomputations of Inverse Trigonometric Functions Solving Exponential and Logarithmic EquationsĪpplication of Exponential and Logarithmic Functions Section 5: Exponential & Logarithmic Functions Rational Functions and Vertical Asymptotes Intermediate Value Theorem and Polynomial Division Midpoints, Distance, the Pythagorean Theorem, & SlopeĬompleting the Square and the Quadratic Formula Precalculus with Limits Online Course Section 1: Introduction ![]()
0 Comments
Leave a Reply.AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |