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 WXY and Z on the diagram. For our answer choices, A says W is 16 multiplied by the skirt of two, X is 16 multiplied by the skirt of three, Y is 16 multiplied by the skirt of two. And Z is 16. B says W is 16 multiplied by the skirt of two, X is 16 multiplied by the skirt of three, Y is eight multiplied by the skirt of three and Z is eight, C says W is 16 multiplied by the skirt of three, X is 16, Y is 16 multiplied by the skirt of two and Z is 16. And D says W is 16 multiplied by the skirt of three, X is 16, Y equals 16 multiplied by the skirt of three and Z equals eight. Now, what do we know here? Well, we know we're working with two adjacent right triangles and once we're working with right triangles that have known angles and sides, then we can start thinking about the trigonometric ratios. And to help us make sense of the tri trigonometric ratios, we can use our special 30 60 degree triangle because notice that one of our right triangles has a 30 degree angle. And for our special triangle recall that if the, if the angle at the base is 60 degrees and at the top of it is 30 degrees, then its size will be one, the hypotenuse will be two and the length of its height will be the square root of three for our 30 60 triangle. OK. So we can use the channel to help us to get the exact values when we are solving in for the sides W and X. So let's go ahead and use that idea. OK. So let's call our side our triangle with a 30 degree angle triangle. A. So in triangle, A let's solve for W. So solving for a side, W notice that in triangle A, we know the value of the hypotenuse is units, we also know that W is the side that's adjacent to the ango, we know adjacent to the 30 degree angle. And that means X is opposite to the 30 degree angle. So if we're solving for W the adjacent side and we already know the hypo news is 32 units, what ratio relates the adjacent side to the high party news? Well, it would have been the core sign. So in solving for W recall that the cosine of an angle equals the opposite side sorry, the adjacent side two, the high party news, in this case, we're dealing our angle is 30 degrees. Our adjacent side is W and our hypotenuse is 32 units. Now, do we know what the cosine of 30 is? Let's use our unit triangle, our 30 60 triangle to help us. Now, if we're talking about the core sign of 30 degrees, the cosine, as we said equals the adjacent side on our 30 60 triangle, that's a skirt of three divided by the hypos, which on that special triangle is two. So the exact value of the cosine of 30 is the square of three divided by two. And now we can use that idea to solve for W because if we multiply both sides by 32 this tells us that W equals 32 multiplied by the square of three divided by two. And we can factor two from the numerator and denominator 22 goes one time, 20 to 32 goes 16 times. So that means W equals 16, multiplied by the square root of three. So now we have a value for W OK, know that we have solved for W let's go ahead and try to solve for X. So solving for X, OK, solving for X and I'm gonna put, solving for double, let me put a line here to separate it. Solving for X. What do we know? Well, X is the opposite side and we already know the high part is units. So what trigger ratio relates the opposite to the high part? Well, it would have been the sign because the sign of an angle equals the opposite side to the adjacent side. So in this case, the sign of our angle, 30 degrees equals the opposite side X to the hypotenuse which is 32 units. So if we figure out the sign of 30 we can solve for X. What's the sign of 30 degrees? Let's go back to our special triangle in our special triangle. The sign of 30 degrees, as we said, the sign equals the opposite sign which is one divided by the hypotenuse, which is two. So the sign of 30 is a half, which then tells us that a half equals X divided by 32. So let's think about it. The sulfa X, we need to figure out what value can we put here? What value divided by 32 equals a half. Well, this tells us that X has to be 16 units because divided by 32 equals a half. Therefore, X equals 16 know that we have values for W and X that means we can solve for Y and Z in triangle B, let's call it triangle B. The triangle is a 45 degree angle. Now, we're looking at a separate triangle, we're going to have to change what we call the opposite, the adjacent and the hypotenuse because now we're dealing with a different angle. OK. So no, right, our adjacent side is Z. OK. Our opposite side is X which we know we know no equals 16. So that's our adjacent. So our opposite side, OK. And our hypotenuse is why? OK. So no, let's, let's first start by solving for why. Why is our hypotenuse? Now, for this triangle with a 4 to 5 degree angle, our unit triangle can help us to get the exact values for the co sign and the sign and the tangent because recall that for a triangle, a 4 45 degree triangle, which is the unit triangle. OK. Then in our unit triangle, 45 degrees here, then let me get my eraser take off this extra bit. Our base would be one unit, our height would be one unit and our hypotenuse would be the square of two. And we can use that to get our exact values for our 45 degree angle for whichever ratio we're talking about. So that means no in solving for why? OK. Solving for why? Right? When we look at why, why is the hypo of our triangle? We know the side opposite to 45 degrees six is units. So what ratio links the opposite and the hypos? Well, we already know that it's the same ratio. So we can use that idea because that idea tells us then that the sign of 45 degrees will be equal to the opposite side, you know, 16 divided by the hypotenuse, which is why now, what's the sign of 45 degrees? Well, from our special triangle recall that the sign of an angle equals its opposite side. 12, the hypotenuse which is the square of two. So therefore, that tells us that the sign of 45 degrees equals one divided by the square of two. So that means I can rewrite it in my equation as one divided by the square of two equals divided by Y. Let's do some, let's cross multiply here. So that means why, OK, why would be equal to multiplied by the square root of two? So that would be our value for a side. Why? No, we're almost there. All we have left is to solve for side that the solving for that. So now let's go back to our triangle. And for our triangle B side Z is the adjacent side, we know the opposite side is 16. And we know that the tangent relates the opposite side to the adjacent side. So the tangent of our angle, so the tangent of 45 degrees equals the opposite side, which is 16 to the adjacent side, which is Z. So if we know the tangent of 45 we can solve, let's go back to our diagram here. What's the tangent of 45 degrees? Well recall that the tangent equals the opposite side to the angle which is one divided by the adjacent side which is one. So no, that means the tangent of 45 is one divided by one which equals one. So we can substitute that value serves as that one equals 16 divided by Z. Therefore, Z must be equal to 16 because only 16, divided by 16 gives us one. No, we have sold for YZXNWX is 16 units. W is 16 multiplied by the square of three, Y is 16 multiplied by the square of two and Z is also 16 units. When we look back on our answer choices that would have been answer choice. C Thanks a lot for watching everyone. I hope this video helped.

Find the exact value of the variables in each figure.

Key Concepts

Trigonometric Ratios

Solving for Unknowns in Triangles

Exact Values of Special Angles

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