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The figure shows a right triangular prism.
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Find the angle between ๐ด๐น and ๐ด๐ถ, giving your answer to two decimal places.
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The first thing we can do in this question is identify our two lengths ๐ด๐น and ๐ด๐ถ.
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The length ๐ด๐น will cut across the rectangular face here.
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The length ๐ด๐ถ will be the diagonal of the base of this right triangular prism.
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The angle between them will be the angle ๐น๐ด๐ถ created here.
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And we can define this as the angle ๐.
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We can create a triangle ๐ด๐น๐ถ in order to help us calculate our unknown angle ๐.
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Letโs take a closer look at this triangle ๐ด๐น๐ถ.
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We know that the length ๐น๐ถ is four centimeters and our angle here is ๐ at ๐น๐ด๐ถ.
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In order to apply trigonometry in this triangle, we need to be sure if we have a right triangle.
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Weโre told in the question that this is a right triangular prism, which means that we have a right angle here at angle ๐ธ๐ต๐ด and at ๐น๐ถ๐ท.
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And since this plane on the bottom, ๐ต๐ด๐ท๐ถ, meets our other plane ๐ธ๐ต๐ถ๐น at right angles, then we know that we have a right angle here at ๐น๐ถ๐ด.
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We note, however, that as we look at our triangle ๐ด๐น๐ถ, we donโt quite have enough information.
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Weโre going to need to find the length of one of these other two sides.
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If, for example, we look at ๐ด๐ถ, we can see on our diagram that this length ๐ด๐ท, which is eight centimeters, is different to the length of ๐ด๐ถ.
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Weโll need to create another right triangle in two dimensions to help us find the length of this side ๐ด๐ถ.
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We can, in fact, create this triangle ๐ด๐ถ๐ท, which will have a right angle at angle ๐ด๐ท๐ถ.
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We can draw out triangle ๐ด๐ถ๐ท on the bottom of our right triangular prism.
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We can see that ๐ถ๐ท is three centimeters and ๐ด๐ท is eight centimeters.
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Donโt worry if your diagrams arenโt perfectly accurate; they donโt have to be to scale.
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Theyโre just there to help us visualize the problem.
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Remember why weโre doing this.
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Weโre trying to find the length ๐ด๐ถ, which is common to both triangles.
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We can define this as anything, but letโs call it the letter ๐ฅ.
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When we find this length ๐ฅ on our first triangle, we can fill in the information into our second triangle.
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As we have a right triangle and two known sides and one unknown side, we can apply the Pythagorean theorem, which tells us that the square of the hypotenuse is equal to the sum of the squares on the other two sides.
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So we take our Pythagorean theorem, often written as ๐ squared equals ๐ squared plus ๐ squared.
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The hypotenuse, ๐, is our length ๐ฅ.
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So weโll have ๐ฅ squared equals three squared plus eight squared.
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Thatโs the length of our two other sides.
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And it doesnโt matter which way round we write those.
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We can evaluate three squared is nine, eight squared is 64, and adding those gives us ๐ฅ squared equals 73.
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To find ๐ฅ, we take the square root of both sides of our equation.
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So we have ๐ฅ equals the square root of 73.
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Itโs very tempting at this point to pick up our calculator and find a decimal answer for the square root of 73.
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But as we havenโt finished with this value, weโre going to keep it in this square root form.
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Now that we have found ๐ฅ, that means weโve found our length of ๐ด๐ถ.
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And so we can use this to find our angle ๐.
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As weโre interested in the angle here, that means weโre not going to use the Pythagorean theorem again, but weโll need to use some trigonometry.
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In order to work out which of the sine, cosine, or tangent ratios we need, weโll need to look at the sides that we have.
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The length ๐น๐ถ is opposite our angle ๐.
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๐ด๐ถ is adjacent to the angle.
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And the hypotenuse is always the longest side.
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Now, weโre not given the hypotenuse, and weโre not interested in calculating it, so we can remove it from this problem.
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Using SOH CAH TOA can be useful to help us figure out which ratio we want.
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We have the opposite and the adjacent sides, so that means that weโre going to use the tan or tangent ratio. tan of ๐ is given by the opposite over the adjacent sides.
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We now fill in the values that we have.
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The opposite length is four centimeters, and the adjacent length is given by root 73.
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So tan of ๐ is equal to four over root 73.
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In order to find ๐ by itself, we need the inverse operation to tan.
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And thatโs finding the inverse tan.
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This function, written as tan with a superscript negative one, can usually be found on our calculator above the tan button.
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Using our calculator to evaluate this will give us ๐ equals 25.0873 and so on.
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And the units here will be degrees as, of course, this is an angle, not a length.
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Weโre asked to round our answer to two decimal places.
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So that means we check our third decimal digit to see if itโs five or more.
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And as it is, then our answer is given as 25.09 degrees.
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And so the angle between ๐ด๐น and ๐ด๐ถ is 25.09 degrees.
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Before we finish with this question, letโs just review what we could have done instead.
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When we started this question, we had this large triangle ๐ด๐น๐ถ which cut through our triangular prism.
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We were told that ๐น๐ถ was four centimeters.
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And we worked out this length of ๐ด๐ถ.
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But could we have done it by working out the length of ๐ด๐น instead?
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If we look at our triangular prism, in order to work out the length of ๐ด๐น, weโd need another triangle.
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We do have a right triangle here.
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And this length of ๐ธ๐น will be eight centimeters.
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However, if we were trying to work out this length of ๐ด๐น, weโd also need to work out this length of ๐ด๐ธ.
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In order to find ๐ด๐ธ, weโd need to create yet another right triangle.
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We would know that ๐ด๐ต is four centimeters and ๐ด๐ต is also three centimeters.
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So we would eventually get the correct answer for ๐ด๐น and, therefore, our angle ๐.
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Itโs just that that second method would involve three triangles instead of the two triangles that we used.