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(0.5 Mark Question (2) If sin x = and sec y where x and y lie between 0 and T/2, evaluate the expression:(a) sin(x + y)(b) sin(2y)...

Question

(0.5 Mark Question (2) If sin x = and sec y where x and y lie between 0 and T/2, evaluate the expression:(a) sin(x + y)(b) sin(2y)

(0.5 Mark Question (2) If sin x = and sec y where x and y lie between 0 and T/2, evaluate the expression: (a) sin(x + y) (b) sin(2y)



Answers

(a) Use the formulas for $\cos (A+B)$ and $\cos (A-B)$ to show that
$$\sin A \sin B=\frac{1}{2}[\cos (A-B)-\cos (A+B)]$$
(b) Use part (a) to evaluate $\int \sin 5 x \sin 2 x d x.$

The half of the number 1 95 in which one has given us sign. Universe. Cossacks. Less cause universe syntax. Let's cause inverse. Synnex. We need to find everybody. X. So uh you know that sign universe. Ex placed course. Universe sex equal to pivot to this huddle. This hardly matters here because you'll be just being like fine words, concepts that Scotland will sign it. So let us right to them. Uh the farmer sign. So this is sign. Universe signed by by two minus theta. And private to Minnesota is always caused theater. Let's cause inverse. Cause bye bye to minus theta. So this is by the way to minus theta. Plus by by two minus theta By -2. Theatre. Now let us or differentiate. So this becomes equal to the web. I This is X. Not data X. There's a complimentary angle formula D whereby the X will be equal to. I'm sorry for this. There's there's there's Differential of the 0 -2. So -2 should be the correct answer and option number A is the correct option. Thank you.

The half of the number 1 95 in which one has given us sign. Universe. Cossacks. Less cause universe syntax. Let's cause inverse. Synnex. We need to find everybody. X. So uh you know that sign universe. Ex placed course. Universe sex equal to pivot to this huddle. This hardly matters here because you'll be just being like fine words, concepts that Scotland will sign it. So let us right to them. Uh the farmer sign. So this is sign. Universe signed by by two minus theta. And private to Minnesota is always caused theater. Let's cause inverse. Cause bye bye to minus theta. So this is by the way to minus theta. Plus by by two minus theta by minus two. Theater. Now let us or differentiate. So this becomes equal to the web. I This is X. Not data X. There's a complimentary angle formula D whereby the X will be equal to. I'm sorry for this. There's there's there's differentiation of the zero minus two. So minus two should be the correct answer and option number A is the correct option. Thank you.

Okay, We have the seeking of access equal to negative, for We were trying to find the half angle identities for sign co sign and now each part. Half angle identities only require that we find our coastline. So and we know that one over co sign that's is equal to this he can next. So we know that our C can is equal negative for So if we solve for Kusenov, X will get cause and of X is equal to negative 1/4. It's just simply flipping our coastline function and therefore are negative for So now we'll start with 1/2 angle identity of co sign X has it's gonna be equal to plus or minus square root one plus coastline ex all over, too. Now, next thing we'll do is from the question. We know that our initial angle is in the second quadrant. That half of our initial angle, if it's in the second quadrant, is gonna be in the first quadrant. So are signed for co sign is gonna be positive. You could have one plus posted of X, which we already found to be negative one force all divided by two. Now we keep solving. We get square root 3/4 divided by two, which is just equal. Positive radical. 3/8. No. So we don't have a radical in the denominator. We can solve this a little further. Get radical 3/2 tens radical, too. And then multiply it by radical. Too radical, too to get radical. Six all over before radical to tens. Radical to is to and then two times two is four. So there's our co sign. Our sign for half angle identity is pretty similar equal to the square root, plus or minus. Still one minus side effects this time all divided by two For the same reason we know where it half angle is gonna be in the first quadrant. So our sign is gonna be positive then We know our co sign is negative 1/4 so we can just look that in. We continue to see at one minus and negative 1/4 over to is gonna be equal 5/8 Then if we don't want a radical in the denominator, we can see that fire sign. Except for two. It's gonna be equal. The radical five divided by 2 10 drag to which were multiplied by a radical. Too radical, too, is ready. 10 over four. There's our half angle for sign and lastly, we have tangents. People plus reminds once again square of one minus co sign of X, divided by one plus co sign that's gonna be equal to. And then we know that our half angle in the first quadrant it's gonna be positive, will be equals. Radical one minus two Co Sativex, which is making one for us, provided by radical one plus negative force is equal to we put under one radical. Now we get five for us, God by 3/4 but the denominators cancel. So we're left with beautiful five over radical three. Then we don't want to radical the denominator. It just leaves us with an X over two people. It's a radical 15 over three, so we left with our final answer. 10 over half angle is radical. 15/3. The sign of our half angle. His radical 10/4. Then our co sign is radical. 6/4

Were given that Sine X is equal to 3/5, where X is an angle in between zero and pi over to We're trying to find Cho sex. So let's draw this out in standard position. We know that it's in quadrant one based on the domain restriction. And if we were to form a triangle with this angle X, we would have opposite as three. And the high part new says five. So Ah, we should find the last sight of the triangle. And we do that by applying Pythagorean theorem. So ah, we'll call this. Ah, Zed three squared plus zed squared equals five squared Zed squared would be equal to 25 minus nine and said would be equal to four square root of 16. So we know that this last side is equal to four. Which means co sign of this angle would be adjacent for over the high part. News five coast to ex. Well, we've got three choices. We got co square theatre minus sign square Theatre to CO square theatre minus one or one minus two. Science where theta. It doesn't matter which one you choose. I'm gonna choose the last option here. This is the same thing as one minus two sine squared X, which is the same thing as one minus two sine squared X is the same thing as Sign X All squared. So this is 3/5 squared. So one minus two times 9/25 which is one minus 18/25 25 minus 18 will give us seven over 25. So that's our answer for part beat part. See? Signed. Two ex. Well, that's the easy one. We only have one formula to pick from. So that's to sign ex co sex. We have sign X. That's two times 3/5 and CO sex. We have from part. Hey, let's 4/5. So that will give us, um 6 24/25


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