SOHCAHTOA |
IDENTITIES |
SOHCAHTOA |
IDENTITIES |
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By using the Law of Sine to solve for side "a". Angle C, which is where the skiiers are, will be our bridge to solve for sides "a' and side "b". Sin 70/ 100 = Sin 45/ a, we can cross multiply to give us a(sin 70) = 100(sin 450). Then we divide sin 70 to both sides to cancel sin 70 on one side so that it will be a = 100(sin 45)/ sin 70. Station A is 75.2 miles away. |
As said before, sin 70/ 100 is our next bridge and it will be used to fine side "b". So sin 70/100 = sin 65/b, we cross multiply to give us b(sin 70) = 100(sin 65). We then divide sin 70 to both sides so that sin 70 cancels on one side so that we can have b = 100(sin 65)/ sin 70. Station B is 96.4 miles away from the skiiers. Once they got rescued and taken back to Police station A they are taken back home. David and Thomas both leave the station at the same time, they diverge an angle of 100 degrees. If David is 4 miles away from the station and Thomas is 5.5 miles away, then how far away are they from each other? |
This picture hopefully sums up the rest of the work. We will use the Law of Cosine to find the missing side. Make sure all the work is right up to this point! Then you write your equation as a^2 = 4^2 + 5.5^2 - 2(4)(5.5) cos 100. After that, we just plug that in to our calculator and we get our answer of 7.3 miles. David and Thomas are both 7.3 miles apart. CONGRATULATIONS!!:D You have solved the problem {=~^0^~=} |
Here we are given a triangle the A,B, and C as their angles and a, b, and c as their sides we can just split the triangle in half to form two triangles. |
Lets focus on angle A, angle C, side a, side c and the height, h, I can show you how they can equal each other. If we take the sine of angle A, it will be h/c but since this is a part of a triangle we don't know what h is. So c is then multiplied to both sides and it will give us h = c(sin of angle A). The same thing will happen if we do the sine of C, except it's h/a and we multiply a to both sides and we'll get h= a(sine of angle C). We can get the height in two ways but in the end the height will be the same for the both of them. Since c(sine of angle A) = a(sine of angle C), and we cross multiply, then ( sine of angle A)/a = (sine of angle C/c.
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It doesn't matter which side is a or b because both sides are the same. We plug it into the Pythagorean theorem and we should have 1^2 + 1^2 = c^2. We square the 1's and add them and we get 2 on that side. Now it's 2 = c^2, we square root each side making 2 into radical 2 and c^2 to c. So radical 2 will be our hypotenuse for the 45-45-90 triangle. Almost done c: but we need to plug in "n" to both sides, so the vertical and the horizontal sides will be "n" and "n-radical-2" for the hypotenuse side. |
Here we have an equilateral triangle ( meaning that all sides are equal) with each side length of 1 and each angle being 60 degrees. To get a 30-60-90 triangle we split the triangle in half down the middle, and get two of them but in this case we are only going to use the one highlighted in pink. The hypotenuse is already there so it's length is 1, the horizontal is also there but it's not 1 because we split it in half so it's actually 1/2. The third side will be found by using the Pythagorean theorem. |
In this case, side "a" will be the horizontal side and side "b" will be the vertical side. When we plug it into the Pythagorean theorem, 1/2 will be squared and it will be 1/4 and 1^2 will just be 1. So we then subtract 1/4 to both sides and end up with b^2 = 3/4 but we need to square root both side to make b^2 to just b. When we square root 3/4 it applies to the top and the bottom, which means that 3 will be radical-3 and 4 will be 2 since the square root of 4 is 2. Our final answer for side"b" will be radical 3/2. |
This is how a 30-60-90 triangle should look like, the hypotenuse side being 2n, side "a" being n, and side "b" being n-radical-3. Well the values are supposed to be radical-3 for side "b", 1/2 for side "a", and 1 for the hypotenuse. To get it to be the derived pattern we just multiply 2 to each side, so the hypotenuse will be 2, side "a" 1 because the 2's cancel each other, and radical-3 for side "b" because the 2's also cancel each other. The "n's" are put in to show that any number can take it's place.INQUIRY ACTIVITY REFLECTION 1. Something I never noticed before about special right triangles is how they are derived form other shapes like the square and the equilateral triangle. PLus ther amount of logcial reasoning it took to solve these types of problems. 2. Being able to derive these patterns myself aids in my learning because it shows that I know how to derive them and how they came to be a 45-45-90 and 30-60-90 triangle, It also helps me review some of my math skills. |