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Given sin(a) =4/5 and cos(b) =1/3, with a and b both in the interval [0,pi/2)find sin(a-b) and cos (a+b)...

Question

Given sin(a) =4/5 and cos(b) =1/3, with a and b both in the interval [0,pi/2)find sin(a-b) and cos (a+b)

given sin(a) =4/5 and cos(b) =1/3, with a and b both in the interval [0,pi/2)find sin(a-b) and cos (a+b)



Answers

Given $\sin (a)=\frac{4}{5}$ and $\cos (b)=\frac{1}{3},$ with $a$ and $b$ both in the interval $\left[0, \frac{\pi}{2}\right):$ a. Find $\sin (a-b)$ b. Find $\cos (a+b)$

Hello. Yeah. So the question belongs to treat no metric identities. If and given that if A N B. Both lying in the first squadron and sign is equal to four by five and Coast B is equal to 12 by 13. Evaluate each of the following identities. So first let me write a statement sign is equal to four by five and Cosby Is equal to 12 x 13. Then we need to evaluate the identities. Of course A minus B. So let me evaluate identities. One by one. Of course A minus B is equal to posey sign, B plus sign who was off B. Since uh A N B both lying enough was squadron. So all of the trigonometry functions are positive. Sign is four by five. Then cozy is equal to square root. We know that co square is equal to one minus sine square, so that will be one minus sine square 18 16 by 25 Which is equal to three x 5. Similarly, we can evaluate the value sign of Let me initially against this, we can write it here or sign B is equal to squared one minus go square B. And that function is positive because be like in the be lying in the first quarter, so that will be one minus 1 44/1 69 that is equal to five by 13. Okay, let me substitute the value. Please value in the expression we get goes A minus B is equal, whose is three x 5 signed The east five by 13 Plus sign is four x 5 and Cosby's 12 x 30. So let me solve it. We get 15 or 65 Plus 48/65. And that is equal to 63 or 65 which is the required value of the function. And the second function is sign A last week saying A plus B is equal to sign. Of course we plus coursey signed the and the value of sign is four by five and for cbs 12 by 13 plus full. She's three by five. And sign bees five by 13. Okay. And that is equal to 48 by 65 Plus 15 over 65. So let me check once and I don't know why I'm getting the same answer. Of course. Oh my God. They dissolved Jose minus base course A course B. And sign sign. So before that. So for that mistake sign Lee. So that will because a. And Cosby is 12 x 32. Let me erase this and it is both of this And that is 12 x 13. Okay. And chinese for by five and sign B. S. Five by 13. Okay So that is equal to 36. Let me raise this also. 36. 20 angry. Okay. 36 plus 20 Okay. And 36 plus 2056 by 65. And the second parties 63/65. Okay. And other parties. Because of the way. Yeah cordis Cause off. We know that because of the ways cool. Squarely minus sine square. And we know that pose pose of A is equal to three by five suppose where is three by five. That is nine by 25. And science where is four by five. So four by five square E. Science very So that will be 16 or 25 when that eventually gives -7/25 and fought by peace. Sign off the way we know their sign off the waist who sign horsey. And the value of sign is four x 5 And go say three x 5. So that eventually called 24 25. So these are the required values of this expression. So hope this created out

Sine a equals water for you, it implies for you it was city. What? And fine. Because be he was 1/3. It implied sewing bee equals. Who's here? Over three in you Go to bye bye. Now we have kind of a new sign. A minus Bi sign a minus bi portal sign. We'll be minus for the air sign. What sign is? Well, don't want to buy you. And also being run over minus for the three over and signed B scale who were to be the women. So just as this is for over 15 minus six. Scared about what? So if we did to over 15 Pirmin were left red Tu minus three scared Do it. Is that require? Now we have to find but it Let's be Let's be a little more Cosby minus sign Sign The board is your If I and course B is one over my no sign is Or what about you? There is a sign be is tools work the bigger chance. And this is who you are. No. Five minus its who What 50 Which is the required

Hello, we have another number 25 and this we need to find sign A plus B because they might be okay and given is sane a sequel to to buy three and Cause be equal to -1 x four and in the interval by went to Dubai, so this is in second quadrant, so if it is in second quarter in and we both had second quarter, so signing A should be positive, Cosby should be negative. Now we need to find the value of course because oh this should be negative Because it is in second quadrant and one science Choir according to what about an identity minus one minus To buy three old square minus 1 -4 x nine, so this is minus Under five x 3. Similarly here we were finding out Sign be since being in second quarter and so sign has to be positive, so plus one minus causes square B, so one minus -1 x four Whole Square, So 1 -1 x 16 Under 15 x four. We have all day, we have completed the required things. So let us get started saying A plus B which is sign a Cosby place cause a Sandy. So we have signed equal to two x 3 Into cost. Be aged under 15 x four. No minus on my phone given all day -1 x four class. Cause a -15 x three signed B Under Road 15 x four. So this is -2 x 12 75 90 503 x 12. So sign a plus B will be equal to -2 -503 by Okay. But be it Because a -1 of course A minus B which is equal to cause a Cosby. MS. Sign a Sandy because a Cosby is because it is uh -005 x three. Cosby Is -1 x four. This saying, is To buy three and signed B is UNDER 15 x four. So this will become minus under fire. I'm sorry, plus under fire because minus and mines were looking place under five plus two Under Rule 15 by for so they should be the answer. These are the answers. Thank you.

We have that. The sign of some angle A is 4/5 and the coast sign of some angle B is 1/3. Now. Both of these angles fall between zero and pi over two. Let's figure out a couple of different things being the sign of a minus bi and the co sign of a plus B. Now we've actually got some identities that could give us these things. That is, we know from the book that the sign of a minus bi well, that's just equal to sign of a Times Co sign of Be Good. So far, we know what both of those things are. Minus co sign of a times sign of be. Well, that's a problem. We don't know Costa innovate or sign of B. Additionally, for our co sign of a plus B, well, that's equal to co sign a co signed be again from identities from the chapter minus Sign A signed Be. So that's a bit of a problem. We don't know Sign B and we don't know co sign a, but there's a way we can figure them out. So let's get to that now. We know that on the unit circle. We have zero right here and pi over two right here, and our A and B are going to fall between zero and pie or two. Thus, we can re draw just that quadrant of the unit circle that is the upper right quadrant 101. So for some angle A, let's say this is angle A. We have that the sign is 4/5. As you know, from introductory Trig, the sign is able to the opposite side over the high pot news. Thus this site is four, and the high pot news is five. All right now, let's just figure out the other side. Well, there's a certain well known triangle. It's a right triangle known as a 345 triangle, meaning that this side must be three. Therefore, we can now determine the coastline of a well. This is going to be adjacent side over high pot news or 3/5. All right, that's one thing figured out. Now let's find the other. Once again, we can draw this triangle, so this time it will be angle B. Now, co sign of B is 1/3. That's going to be adjacent overhype odd news now we can figure out what are missing. Site is so, as you know, for the Pythagorean theorem, we have a squared plus B squared equals C squared or, in this case, one squared plus X squared equals three squared. Giving us X squared equals nine minus one, which is eight. That's moving the one to the other side with subtraction and therefore acts equals the square root of eight. So therefore looking at this triangle were able to determine that the sign of Angle B is going to be opposite over high pot news or square root of eight overhype. Odd news Square debate over three. And now, with this in mind, we can plug everything in and figure out what we were looking for. Let's start out with the sign of a minus bi. So just parsing this off here. All right, so that's sign of eight Times Co sign of B or 4/5 times 1/3. Those were given The problem minus coastline of a time sign of B Coastline of A was 3/5 times Sign of B was squared of 8/3, and now we just have to simplify. So this first term Ah, the four times one just gives us 45 times three is 15. So 4/15 minus. Then these threes can cancel out, leaving us with just squared of eight over five. And that is the sign of a minus bi. All right, that's one out of the way. Now let's find the other. That is the co sign of a plus B. So with this one I wrote down the identity above. We have to have the two coastlines multiplied together. That is coastline eight times co sign be eso co sign a. We determined that was equal to 3/5 and then times Coastline B, which was given as 1/3 minus the product of the sign terms. That is, Sign A, which is 4/5 times sine b, which is squared of 8/3. And now we can multiply this out. This will give us eso these three times one will multiply the equal three divided by five times three is 15 minus four times the square root of eight, also over 15 and we can actually simplify this four times squared of eight and put this over one fraction bar giving us three minus so squared of eight that is squared of four times to so, Taking the square to four out, we get a two months right out front two times four is eight square to all divided by 15 and that is the co sign of a plus B, thus using some identities and some clever unit circle tricks were able to determine what sign of a minus bi in Khost out of a plus B R without actually figure out what A and B are.


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