Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Question

Answered step-by-step

The German Mathematician Karl Weierstrass (1815-1897) noticed that the substitution $ t = \tan (\frac{x}{2}) $ will convert any rational function of $ \sin x $ and $ \cos x $ into an ordinary rational function of $ t $.(a) If $ t = \tan (\frac{x}{2}) $ , $ -\pi < x < \pi $ , sketch a right triangle or use trigonometric identities to show that$ \cos \left (\dfrac{x}{2} \right) = \dfrac{1}{\sqrt{1 + t^2}} $ and $ \sin \left (\dfrac{x}{2} \right) = \dfrac{t}{\sqrt{1 + t^2}} $(b) Show that$ \cos x = \dfrac{1 - t^2}{1 + t^2} $ and $ \sin x = \dfrac{2t}{1 + t^2} $(c) Show that$$ dx = \frac{2}{1 + t^2}\ dt $$

Video Answer

Solved by verified expert

This problem has been solved!

Try Numerade free for 7 days

Like

Report

Official textbook answer

Video by J Hardin

Numerade Educator

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus 2 / BC

Chapter 7

Techniques of Integration

Section 4

Integration of Rational Functions by Partial Fractions

Integration Techniques

Harvey Mudd College

Baylor University

University of Nottingham

Lectures

01:53

In mathematics, integration is one of the two main operations in calculus, with its inverse, differentiation, being the other. Given a function of a real variable, an antiderivative, integral, or integrand is the function's derivative, with respect to the variable of interest. The integrals of a function are the components of its antiderivative. The definite integral of a function from a to b is the area of the region in the xy-plane that lies between the graph of the function and the x-axis, above the x-axis, or below the x-axis. The indefinite integral of a function is an antiderivative of the function, and can be used to find the original function when given the derivative. The definite integral of a function is a single-valued function on a given interval. It can be computed by evaluating the definite integral of a function at every x in the domain of the function, then adding the results together.

27:53

In mathematics, a technique is a method or formula for solving a problem. Techniques are often used in mathematics, physics, economics, and computer science.

13:43

The German Mathematician K…

03:43

The German mathematician K…

04:25

01:06

In calculus, trigonometric…

01:14

01:32

01:23

01:15

00:59

08:46

An integrand with trigonom…

01:20

we were given that he equals tangent events over too. And we'LL go ahead and show these identities for co sign and signing party. So using this fact here, that's good. And draw a triangle, suffer a and then hear this This picture is kind of using the fact that are we can draw something like this. So we need to use the fact that X is between negative pion pie. And really I should come back over here on ankles X over too. So the fact we're this being used, this is negative pirates who, less than ex Western pirates who so we don't have it is not bigger than the right angle. Okay, so then if he equals tangent about supper too, so we can also write that extension of this angle that we just drew equals t divided by one. So a tangent is opposite over adjacent, and then we could use the potential really there and so find hypos. Now we have all three sides so we could find coastline inside. So looking at co sign, this is just the Jason over iPod news. So we get one over the square roof. He's fair plus one So that's this formula up here. Check now for the second sign. This is opposite over hypotenuse so t over the square of T Square plus one. That's what the formula here says. So that verifies the second formula. So now we want to verify and part B these formulas for co sign next and sine X. So to do this, let's go ahead and use part A. So let me go to the next page here. So for part B, let's use the formulas that we just found the party, eh? One over. He squared, plus one of the radical and then sign t over teeth. Where? Plus one in the radical. So now, for example, if you want to find Cho Side of X, you could write This is co sign of two Times X over, too. And then you could use your double angle formula for co sign So co sign of two y, for example, is co sign squared y minus sine squared away. So here we're used this formula, but with y equals, except for two. So this becomes co sign squared events over too, minus science where that's over, too, and now, using our formulas from party, eh? This is one over T Square plus one, and that whole quantity is square minus sign of X over too. And we'll also square this entire expression and then we just go ahead and simplify. So have one up top minus He squared also on top And then when we square that radical, we just get t squared plus one. And that's the formula that they wanted for co sign of X Now we'LL go to the next page and we'll do the same thing for sign two times X over too. Now we use the fact sign of two. Why could be written is to sign Why cosign? Why? This is the double angle formula for science. So using that here with why is ex over too? We have to sign. There's our why and that co sign what now? Using the formulas from party, we found sign and then for co sign also from party here and then simplify that combined those radicals. There's a party now let me go to the last place for party. So recall the definition of teeth. Let's go ahead and take a differential on each side and then by the chain rule. We have a one half and then DX I could also write This is one plus. Can square eggs over to this is by the protection identity and just write that too in the denominator using teeth. The definition this is one plus t squared over two and then just go ahead and take this equation here and saw for DX. And we have two over one plus t square dt. And that verifies the formula for D s. And that's our answer.

View More Answers From This Book

Find Another Textbook

03:11

Find the values 0f X Y, and z The diagram is not drawn t0 scale38819…