Math Mistakes: Law of Sines

Link to Math Mistakes↓

http://mathmistakes.org/?p=933 

What is wrong?

The mistake they did was when solving for the law of sines, they did not divide by the side corresponding to the angle. Instead, they only cross multiplied and did not divide to get sine B by itself. They left the decimal with sine of B instead of getting sine B on its own because that is what we are solving for. As a result, they got the wrong value for <B.

What they should have done:
When we are solving for the law of sines, we set up our magic number and our bridge. The angles must correspond to the sides. While this person set up the problem correctly, they solved it the wrong way. When you cross multiply you get 4.21sin30.4=1. 43sinB then, you divide by 1.43 to get sinB=1.49.  We should have taken the inverse sine but we notice that it is greater than one and it is not possible.

 

WPP #12: Unit O Concept 10

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WPP #11: Unit O Concept 9

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Derive the Unit Circle

Special Right Triangles

 To start off, this is a 30°, 60°, 90° right triangle. What do we know about this specific right triangle? Well, we know that the leg opposite 30° is n, the leg opposite the 60° is n√3, and the hypotenuse is 2n. We want r to be 1. So, we divide by 2n. And....what we do to one, we do to all the others!! As you can see in the picture on the left, we divided them all by 2n and got our values for r, x, and y.

  

How does it connect to the UNIT CIRCLE?

 Well, if we create a coordinate plane, with the “X” SIDE being the x-axis, we can plot each of the vertices as an ordered pair. We also notice that our triangle is in QUADRANT I of the Unit Circle. The origin is (0, 0). The 90° angle ordered pair would be (√3/2, 0). This is because we go to the right “√3/2” units (our x value) and we are still on the x-axis so our y is 0. The 60° angle ordered pair is (√3/2, 1/2). Since we go right “√3/2” (x value) units and go up “1/2” units (y value).
          *The BIG PICTURE, shows us how the right triangle is in the unit circle and gives us one of our “magic five” values.*

  

More Special Right Triangles

 The same works with 45°, 45°, 90° right triangles!! What do we know about this specific triangle? Well, we know that the leg opposite 45° is n, the other leg opposite the 45° is also n, and the hypotenuse is n√2. We want r to be 1. So, we divide by n√2. And....what we do to one, we do to all the others!! As you can see in the picture on right, we divided them all by n√2 and got our values for r, x, and y.

The same as above works for this triangle! *The BIG PICTURE, shows us how this right triangle is in the unit circle and gives us one of our “magic five” values.*