WPP#13/14: Unit P-Concepts 6 and 7

Photo-credits:

http://24.media.tumblr.com/48a60113d127af3168cd9ff610d86684/tumblr_my0biwlZRg1qmxwggo1_r2_500.jpg

Hazel grace is located 135 miles due south of Augustus Waters. Destiny is calling out for their first meet in order for them to meet and fall in love. Destiny's call indicates that Hazel Grace is located at a bearing of 60 degrees; Destiny's call indicates that Augustus Waters is located at a bearing of 131 degrees. How far are Hazel Grace and Augustus from Destiny?

Photo-credit: lovely, hand-drawn pictures by Mrs. Ventura

Two planes leave the airport at the same time, traveling on courses that have an angle of 125 degrees between them. If the first plane travels at 35 miles per hour and the second travels 28 miles per hour, how apart are the two planes after traveling for 5 hours?

Photo-credit: lovely, hand-drawn pictures by Mrs. Ventura

BQ#1- Concepts 1 and 4

1. Law of sines-Why do we need it? How is it derived from what we already know?

Photo-credits: lovely, hand-drawn pictures by Mrs. Ventura

Using the picture from above, we will go step-by-step of the derivation of the equation. First things first, we must know that the law of sines is needed to solve missing parts of a non-right triangle. A regular right triangle can be easily solved because it has all the needed components, that being, most importantly, a right triangle. So, splitting a non-right triangle we get two right triangles. We solve for both angles A and C using sin. From there we get sinA=h/c and sinC=h/a. We take it one step further and we multiply by c  and a ,for the two separate equations, in order to get h alone. Having h alone for both equations, we then combine them and move on over the a and c to their appropriate sides. From there we get the law of sines. Of course the law of sines is not just restricted to a or c, it also works for b as well. The law of sines fluctuates according to what specific information you are looking for.

Photo-credits: lovely, hand-drawn pictures by Mrs. Ventura

4. Area formulas- How is the "area of an oblique" triangle derived? How does it relate to the area formula that you are familiar with?

The area of an oblique triangle is derived from the area formula and a combination of a trig function. The area formula is A= 1/2bh. With a non-right triangle, trying to solve the area is much more difficult. Therefore, we split the triangle with a perpendicular line and get two triangles. From there, we find angle C. SinC=h/a, we multiply a to the other side in order to get h alone. Moreover, we plug in h into the original area formula. That, is the equation for solving the area of an oblique triangle.

It's similar to the area formula we know, except we manipulate it, in order to get what we need. The only difference is since we do not have h we use our sources and use it to the best of our abilities.

Below are other equations that are used to solve oblique triangles:

Photo-credits: lovely, hand-drawn pictures by Mrs. Ventura

BQ #7- Where does the difference quotient come from?

Photo-credits: lovely, hand-drawn pictures by Victoria

Looking above at the image, we notice that our first points the x-value is x and our y-value is f(x). Technically looking at the graph not a certain distance is given so that's the reason why we use f(x). For the very same reason the next set of points x-value is x+h and the y-value is f(x+h). The sole purpose of this graph is to find the slope tangent line on the two specific points. To find the slope of the tangent line we must use the slope formula which is m=(y2-y1)/(x2-x1). We plug in the points to the formula and continue on simplifying as much as we can: cancelling/combining like terms, simplifying, etc. Once we have finished the whole lovely process we get the difference quotient! Voila! The difference quotient is extremely helpful in finding derivatives for a certain graph. A visual is displayed below for the process of plugging in to the points to the slope formula ---> to becoming the difference quotient.

Photo-credits: lovely, hand-drawn pictures by Victoria