Homework 4 trigonometric ratios & finding missing sides

Download Plane Trigonometry - Homework 4 Solutions | MATH 111 and more Trigonometry Assignments in PDF only on Docsity! MATH 111: HW 4 SOLUTIONS AND COMMENTS 4.1 #3. The Pythagorean theorem states that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In the right triangle of this problem, we therefore have hypotenuse2 = 12 + x2 and so hypotenuse = √ 1 + x2 . 4.1 #4. We know one side of the rectangle has length 4 inches; to find the area of the rectangle, we want find the length of the other side of the rectangle. The diagonal divides the rectangle into two right triangles. Both of those triangles have hypotenuse 5 inches and one side of length 4 inches. By the Pythagorean theorem, 52 = 42 + (other side)2 , so (other side)2 = 25− 16 = 9 and therefore (other side) = 3 inches. Therefore, the rectangle has sides of length 3 inches and 4 inches, and the area of the rectangle is 3 inches times 4 inches: that is, the area is 12 square inches. 4.1 #6. The angles sum to 180 degrees (a straight angle). 4.1 #8. No, there cannot be a right triangle with sides of length 1, 2, 3. If there were such a triangle, the hypotenuse (always the longest side) would have length 3. Then the Pythagorean theorem would require that the equation 12 + 22 = 32 be correct. But, we know that 12 + 22 does not equal 32 (because 1 + 4 does not equal 9!!) So there cannot be a right triangle with those side lengths. 4.1 #15. The first step is to note that 1 mile is 5280 feet. (If you didn’t know this already, this is easy to look up on the internet.) In the problem, the track expands to 5280+2 = 5282 feet. The two halves of the track are each right triangles, with base 5280/2 = 2640 feet and hypotenuse 5282/2 = 2641 feet. The height of the midpoint of the track is therefore height2 = 26412 − 26402 = 5281 and so height = √ 5281 ≈ 72.67 feet . 4.3 #3. The ratio of two sides of a Golden rectangle is φ = 1 + √ 5 2 ≈ 1.618 .. The ratios of the sides of the rectangles in the problem are: 5/3 ≈ 1.666; 11/8.5 ≈ 1.294; 14/11 ≈ 1.273; 17/11 ≈ 1.545. The closest of these to 1.618 is the first, so the index card most closely resembles a Golden rectangle. 4.3 #4. The two equations have the same solutions because they are essentially the same equation: the second equation is gotten from the first equation by taking the reciprocal of both sides. Therefore, any number which solves the first equation will solve the second, and vice-versa. 4.3 #5. Recall the quadratic formula: if we want to solve the equation ax2 + bx + c = 0 for x, then there are two solutions: x = −b±√b2 − 4ac 2a . Now let’s look at the particular equations in this problem. (1) Cross-multiplying yields 2x(x− 1) = 1, which expands to 2x2 − 2x = 1. Bring the 1 to the other side of the equation: 2x2−2x−1 = 0. Now apply the quadratic formula (in this case a = 2, b = −2, c = −1) to see that the two solutions are x = 2±√12 4 = 1±√3 2 . Caution! Notice that √ 12 = √ 4 · 3 = √4 · √3 = 2√3; so it is NOT the case that√ 12/4 = √ 3. (2) Cross-multiplying yields x(x− 4) = 2 · 3, which expands to x2 − 4x = 6. Bring the 6 to the other side of the equation: x2− 4x− 6 = 0. Now apply the quadratic formula (in this case a = 1, b = −4, c = −6) to see that the two solutions are x = 4±√40 2 = 2± √ 10 . (3) Cross-multiplying yields 3x(x + 1) = 2, which expands to 3x2 + 3x = 2. Bring the 2 to the other side of the equation: 3x2 +3x−2 = 0. Now apply the quadratic formula (in this case a = 3, b = 3, c = −2) to see that the two solutions are x = −3±√33 6 . 4.3 #9. Yes, the new smaller rectangle is a Golden rectangle. The sides of the new rectangle are half the length of the sides of original rectangle; but this means that the ratio of the sides of the new rectangle is exactly the same as the ratio of the sides of the original rectangle.

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I am given a preliminary to find out what X, Y and Z are. Well, right? There is a problem here. Let's get to know some basic concepts. There's a triangle straw and a soybean hard riding it. This is the first time I think this is stable and his is 90. That's right. Science theater is equal to oppose Dennis the worst thing to do next to hypotenuse and then theater they called to oppose it over at justice. The cinematic ratios are going to be used to find a solution to the problem. We've got a problem trying to see so why not X. That's right. Where do you apply if you want to say "Do you go four years time theatre sign 30 Right 7 30 Let's oppose it". Act is going to be equal to eight trees if they will do acts over Exchange three happiness. I wonder why we need this length in this area. We need to hear this long. Well, right? For that, I'm going to use the same triangle, what I'm going to use is 30 so costs are equal. Maybe the doctor is Cuban, but I understand. We have happiness and it's a B over. External routes 30 and 16 have given us root 2. It's equal to a b. over 16 road treat. There is a A. There is a person named B. It is equal to 16 routes three times. Well, right? All is divided by two. General three times and divided by two. That will have to be 24. So this is 24. I want to find out what happened. The right at this angle is 45. The angles and property of the triangle are 180. We have 45 plus 45 and 90 plus 90, so the single would be 45. I'm debating $9 in property we have used here. This is 45 to 45 plus 45 to 90 in the 90s. The two angles of the triangle are equal in the sight of a positive, so we know that. That will be equal. This will be the reason why this will be equal to Y. So why? That's also the reason? Is equal to Y. Which champ will solve this? We would like to use the pythagorean triangle in crime. We are going to use the despite the beautiful to europe. If we use this term now, we will have wide square, right? There is a white square and y squared. You can use the two White Square e Quilts 24 sq. You start twice for each time. This is where you're going to go to So 12. I will get 12 times 24 for Y squared. Well is two times two times three Times 24 is two times two times three, and y squared is equal to that. Well, right? Why is it that I am going to make a pair too? Time is true times three. And then Route 2. I don't know why someone is coming up to me with a route to take. Why is well road to? Rule two says that these are coming out to be 12 to So I and D. Thank you so much.