Find the exact value of the expression12 cos 135 13 Jlsin sin12 cos 135 13sin sin(Type an exact answer in simplified form )...

Find the exact value of each expression.

$$\cos \left(2 \sin ^{-1}\left(\frac{5}{13}\right)\right)$$

So 10 of sine inverse off 12/13 that will be equal to 2.4.

So sickened off sine inverse of 12/13 is saving, saying one over co sign sine inverse of 12/13. So when you're plugging in things on a calculator, you can do the operations like this since there is no second, so the answer here will be 2.6.

For this video, we will evaluate coastline of sine inverse if I 13th minus Tanja inverse up 3/4. So we're dealing with co sign of the difference between two angles, which means we can use this identity here to aid us. So we start off by writing that this is equal to co sign of the first angle, which is signed in verse of 5/13 times Coastline of the second angle, which is Tanja inverse of 3/4 the minus. We see here, then converts to a plus in this difference formula, and we do sign of the same angles. So we have sign of sine inverse of 5/13 times. Sign of tangent in verse of 3/4. So that's our initial set up with this problem, the next step in this evaluation, we need to go to each of these four different factors we see here and evaluate each and every one of them. So let's start with co sign of sine inverse of 5/13. Our strategy here is to let Fada be equal to sine inverse of 5/13 so that by substitution, we're really dealing with co sign of Fada now back to this equation. If we take sign of both sides, the left hand side will be science data, but we'll have a cancellation on the right hand side, since Sign was sine inverse are inverse functions. So now we know. Scientist Ada is 5/13 to determine CO ST Data will use the Pythagorean identity. That coastline squared of Theta plus ST Square to Fada, which should be 5 13 squared or 25 169th is equal to one or 169 160 nights. If we subtract the Fraction 25 160 nights from both sides, then cosine squared of theta would be 144 169th. Then, if we take the square root of this equation, we get that coastline data is 12 13th. Since we're dealing with inverse functions, we only take the positive route in this set up. So we are one factor down. Co sign of ST Inverse of 5/13 is 12 13th and will multiply that by the result of co sign of Tanja in verse 3/4. Let's work out this evaluation down below us. Well, so we need to evaluate co sign of tension in verse of 3/4. Once again, we're going to let a Nangle let's call this one fee, be equal to tension in verse of 3/4. Then we take tangent of both sides. But first bike substitution we now have that this is equal to co sign of fee. So back to this equation, the left hand side will write as tension fee, which means the right hand side is tangent of Tanja Inverse of 3/4 and through cancellation we obtain just 3/4. Now we'll use the identity that tension squared. Ah, Fi plus one is equal to seek and squared of feet, and second squared is, in a sense, convenient because he can't is the reciprocal of co sign. So we use that fact down the line. Next, let's make a substitution here for attention squared, we know tension is three force, so if we square it, we get 9/16 plus one, which is 16. 16th is seeking squared a fee, so 25 16th is he can't square to fee. If we take the square root, herbal sites will get 5/4 is C can't of feet. Then this equation we see here, if we take the reciprocal both sides, that gives us co sign fee, which is found to be 4/5. So we'll go times 4/5 back up at the original evaluation. Then the next term is going to be added. And when we see sign with sine inverse, we get a freebie. There's an immediate cancellation there, which allows us to carry down 5/13 then that will be multiplied by the result of this evaluation. Sign of Tanja Inverse of 3/4. So let's do that Next down here. Sign of 10 geant inverse of 3/4 notice. We've already named fee to be tension in verse of 3/4 so this is now sign of feet. We also know that co sign fee is 4/5 so it's used the Pythagorean identity that says sine squared a fee plus coa ST Square to fee, which would be 4/5 squared or 16 25th is equal to one or 25 25th if you prefer. If we solve for science word, we obtain that sine squared. A fee will be equal to subtract 16 25th from both sides so 9/5. Then if we take the square root of this equation, we get the positive route. 1025th. There we get the positive route 3/5. So going back up above here we have a 3/5 that carries there. So now we're ready to finish off our evaluation. We can write that this is equal to 48 65th plus 15 65th which sums to altogether 63 65th. And that's the evaluation for this originally expression.

Okay, so I want to use identity sigh of A plus B equals the sign of A. Times close eye of B plus uh plus sign of the times co sign of A. Okay, so we can rewrite this as sign of empire over 12 plus pi over four. Okay, so this equals the sign of 10 point over 12 ft. It's actually five pi over six times. Co signing off Pi over four plus close eye of Uh five pi over six times. Sorry, of Pi over four. Okay, so let me draw a little diagram here. So saying five pi over six is some Place around here. Okay, so this is and go five pi over six. Mhm. So this equals two one half. Okay. Time. Because the impact of pile before he goes to squirt of 2/2 And co sign of five pi over six equals 2 minus squares of 3/2 times squared of 2/2. So I have Squared of two squared of six over four. Okay, so this is the value of this uh this expression