Suppose A = arcsin( −2 5 ) and cos B = 1 4 where 3π 2 < B < 2π. Find the exact value of sin(A − B).

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Okay, we are given with sign and Cosby with a lying between lying in the third quarter. That is 5 to 352 and be just too well by 13. Where B is lying Cosby just 213 and B is lying between 3.222 pilots. His fourth content. Now let's calculate sinus cause A value. So course we can make a triangle here. This is 90 and this is angled A. So sign. Three ties opposite upon hypothesis. We can neglect minus sign for now. And this length can be careful by using pythagoras of photography's. This side will be root off five square minus four squares 25 minus 16 is nine. That gives three. Now you can find cause A. So causes adjacent upon high fruitiness. So three by five. And now we look into the quadrant where is lying. So this is in third quadrant and in third quarter and we have caused negative. This will also be Thank you too. Right, so we are done with cozy. Similarly from this, we can calculate sign B. So again forming a triangle with this angle B. This is 90 degree. So Cosby has given us that is too ill and 13. And now this angle can be calculated using pythagoras, which is root of 13 square minus dual squares for 1 69 minus 1 44. That gives 25 and Rudy's five. So our sign A will be opposite or hyper tennis. And in the fourth quarter in that is three power to to buy sign is negative. So this will be negative. Now to calculate cause A by two we have to use formula course square tita minus sine squared theta is because to theater. Right? So here we can write this as to cause square theater minus one has caused to theater. Now, if the security theater is A. This becomes caused A. This becomes one minus two times cause square A by two. Right? So we have cost value which is sorry this is B. So of course he values minus three by five minus one. And we need this value. So in solving this for cause A by two where you get 1/5 and that's an answer for cost A by two. Now to calculate an A minus B we can apply formula of 10 e minus piece of 10 A minus B. Is an A. My nest on B over one plus 10 times can be now 10 is nothing but signing upon cozy and we know the value so sign upon cause will give minus four by five point minus three by five. Just four by three. And then we will be signed be upon Cosby so minus five by 13 or 12 by 13. So that will be minus five by 12. The same thing here, four by three minus 5, 12. So I'm solving this, we get 63 by 16 and let's find him. And so thank you.

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