Find the exact value of the variables in each figure.

Question

Accepted Answer

Welcome back. Everyone. In this problem, we have a figure that consists of two adjacent right triangles. And we want to determine the exact values of RST and U for our answer choices. A says that R is 11 multiplied by the skirt of three. S is 22 T is 22 multiplied by the skirt of three divided by three and U is 44 multiplied by the skirt of three divided by three. In answer choice, Bs and U remain the same as in answer choice. A but no, R is 11 multiplied by the square of two and T is 22 multiplied by the square of two divided by two. For answer choice, CS remains the same as in choices A and B but R is now 11 multiplied by the skirt of three, T is 11 multiplied by the skirt of three, divided by three and U is 22 multiplied by the skirt of three divided by three. And finally, in choice, Ds and U will remain the same as in choice, C but no, R is 11 multiplied by the square of two and T is multiplied by the square of two divided by two. No, we know that we're dealing with two adjacent right triangles. And since we're dealing with right triangles, we know that we can either use the idea of the trigonometric ratios or the Pythagorean theorem if we know at least two of the sides. Now, what do we know about those? Well recall that for our trigonometric ratios, the sign of an angle equals the opposite side or the ratio of the opposite side to the hypotenuse. The core sign is the ratio of the adjacent side to the hypos. And the tangent is the ratio of the opposite side to the adjacent. Likewise, we also know that we're dealing with a special right triangle. It's actually the right triangle that has 30 60 degree angles. And recall for that type of right triangle. Then that means if we have the Angus 30 degrees 60 degrees and our right angle, our base is one hypotenuse is two and the height is the square root of three. And what this does, it helps us to easily find exact values for any trigonometric ratio uh for 30 or 60 degrees. So that's going to help us while we work. So what we need to do, we can try to find the side one by one RST and U. And then we can use any of these ideas to help us. So let's start by trying to solve for R. Now, if we look at the triangle that R is in, let's first start by labeling the opposite and adjacent and hypotenuse sides for R. OK. So let's stick to the 30 degree angle for no. In our first triangle, let's call this triangle triangle A. Now in the opposite side to the 30 degree angle would have been the side with 11 units and the adjacent side would have been side R and the hypotenuse for this triangle would have been side s so now that we have that we should be able to solve for R and S in solving for R notice that we have one of the angles in the triangle and we have one of the sides. So a trigonometric ratio would make sense, it would make sense to try and solve for R here using that. Now, what side do we have? Well, we have the opposite side and we want to solve for are the adjacent side. So what trigonometric ratio links the opposite and adjacent sides? Yes, it would have been the tangent. So we can solve for R using the tangent for our angle. So solving for R, this tells us that the tangent of 30 degrees equals the opposite side, which is 11 divided by the adjacent side, which would have been R. Therefore, R would be equal to 11 divided by the tangent of 30 degrees. Do we know what the tangent of 30 degrees is? Well from our special unit triangle or our special 30 60 triangle. Rather the tangent of R the sorry, the tangent of 30 sorry would have been one tangent of 30 equals one divided by the square root of three. So therefore, R equals 11 divided by one divided by the square of three, which would be equal to 11 multiplied by the square of three, divided by one, which is 11 multiplied by the square root of three. Therefore, R equals 11 multiplied by the square root of three. Next, let's solve for S so let me, I, I think I'll have to set my problem up differently. So let's start solving for S OK. Now we know that the, so for S S is the hypotenuse of Chiang A, what do we already know? Well, we know the opposite side in a and now we know the adjacent side, but we can use the opposite side to help us. So since we know the opposite side and want the hypo, we need a ratio that links both, which is the sine ratio. So solving for S we can use the sine ratio that says the sign of an anger equals the opposite side. In this case, 11 divided by the hypotenuse which would have been s now, do we know what the sign of 30 degrees is? Let's go back to our special triangle. The sign of 30 degrees equals the side opposite to the 30 degree angle one divided by the hypotenuse which is two. So the sign of 30 is a half. So we can rewrite the sign of 30 as a half. So that means a half equals 11 divided by S. Now, if you look at both sides of the triangle 11 or the equation rather 11 divided by what would have given us a half. Well, s would have to be 22 because 11 divided by equals a half. So this tells us that S equals 22 units. So now that we have solved for R and we have solved for S let's try to solve for T and U. So I'm going to move to a second triangle. Let's call that triangle B. OK. And no, let's work with our 60 degree angle for triangle B. OK. So let's work with our 60 degree angle. And we already know SS is 22 units. Now, for triangle B, the side opposite to the 60 degree angle would have been side s the side adjacent to it would have been side T and the hypotenuse would have been side U. So no, we can solve for T, we know the opposite side and we want to solve for the adjacent side. The ratio that we need is the tangent. So let me draw a line here. OK. Solving for T, OK. The tangent of 60 degrees a tangent of 60 degrees equals the opposite side, which is 22 units. Side S divided by the adjacent side which is T no, we can go ahead and solve first, we need to figure out what the tangent of 60 is, we can't leave out our special triangle. So on our special triangle, the tangent of 60 degrees, OK would be equal to the opposite side in our special triangle, which is the skirt of three divided by the adjacent side, which is one. So in other words, the tangent of 60 is the square root of three. So I can use that value in my equation here because that means the square of three equals 22 divided by T, which tells us then that T equals 22 divided by the square of three. And we can simplify that fraction by rationalizing because when we rationalize, multiplying it by the numerator and the denominator by the square of 3 22 multiplied by the square of three will be in the numerator. And in the denominator, the skirt of three multiplied by the skirt of three equals three. Therefore, side T is 22 multiplied by the skirt of three divided by three. No, we're almost there. The last one we need to solve for is side you. Now let's think about it beside you. What do we know? And what do we need? Well, when we know the opposite side is 22 units, now we know the value of T and we need to solve for the hypo you what ratio links the opposite and the high continuous Well, again, that would be the sign. So we need the sign of 60 degrees. So solving for you note that the sine of degrees equals the opposite side in our triangle, which was 22 divided by the hypotenuse, which was U. So the question is, what is the sign of 60? Again, let's refer to our unit triangle, our special triangle. Sorry. Now, for the 60 degree angle, the sign would have been the opposite side which is the square of three divided by the hypotenuse, which is two. So the sign of 60 is the square of three divided by two. And we can substitute that value here because that no means the square of three divided by two equals 22 divided by U. Therefore, you, if we cross multiply both sides, U multiplied by the squared of three equals 22 multiplied by two, which is 44 which tells us then that U equals 44 divided by the square of three. Again, to simplify, we can rationalize. So it means multiplying the numerator and the denominator by the square of three. And in our enumerator, 44 multiplied by the square root of three equals well, 44 multiplied by the square of three. And in our denominator, the square of three multiplied by the square of three equals three. Therefore, we now have all our values for RST and UR equals 11 multiplied by the square of three S equals 22 T equals 22 multiplied by the skirt of three, divided by three and U equals 44 multiplied by the skirt of three divided by three. When we go back to our list of choices that would have been answer choice. A thanks a lot for watching everyone. You really did. Well, today, I hope this video helped.

Find the exact value of the variables in each figure.

Key Concepts

Trigonometric Ratios

Pythagorean Theorem

Unit Circle

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