WPP #12: Unit O Concept 10: Solving angle of elevation and depression word problems

A) Robert is out skateboarding, filming for his part in his new skate video, and he wants to do a trick off of some store. He is walking towards the store he is going to do a trick off of and finds out that the angle of elevation from where he's standing is 73 degrees. If Robert is 7 feet away from the store, what is the height of the store?

B) Robert is now on top of the store and before doing anything he looks down from the edge of the store and estimates that the angle of depression is 59 degrees. He knows the store is 23 feet high. What would be the distance from the top of the store to the ground?

I/D #2: Unit O Concepts 7-8: How can we derive the patterns for our special right triangles?

Inquiry Summary Activity

1. 30-60-90

   The given triangle is an equilateral triangle and all of its angles are 60 degrees and each side is equal to 1. To get the 30-60-90 triangle you must draw a line down the center of the triangle, leaving one angle being 90 degrees, the other one is 30 degrees, and the other angle is 60 degrees. Like the equilateral triangle these all add up to have a total of 180 degrees. Since the hypotenuse of the triangle remains as 1, we get 1/2 for one of the sides-the x value-since the line was drawn down the center of the triangle and the original number for that side was 1. With this information we use the Pythagorean theorem to find the missing side value. When you solve for the missing side value-the y value- you end up with y^2 equaling radical 3/4 but since the 4 can break down into 2x2 the answer is radical 3/2. Next you multiply each side by 2. When you multiply the angle opposite of 30 degrees you should end up with "n" after multiplying 1/2 by 2, when you multiply the hypotenuse by two you should get "2n" since the value of the hypotenuse was 1, and for the other side you multiply 2 by radical 3/2, the two's cancel out and you're left with radical 3. The "n" values are there to show that the value can keep expanding. 

2. 45-45-90

   We are given a square with all equal sides whose values are all 1. In order for us to find our 45-45-90 triangle we first draw a line that goes from one corner of the square to another. The 90 degree angle is the corner that does not have a line drawn on it and the other two angles are both 45 degrees. Since we have our x and y values for the triangle we use the Pythagorean theorem to solve for the hypotenuse. The hypotenuse ends up being radical 2 and the other two sides remain as 1. Since the value can keep expanding we use "n", thus giving us the x and y sides being "n" and the hypotenuse ends up as "n radical 2." 

Inquiry Activity Reflection

Something I never noticed before about special right triangles is that "n" can expand. 

Being able to derive these patterns myself aids in my learning because I now know why each value is what it is. 

I/D Uint N Concept 7: How Do SRT and UC relate?

Inquiry Activity Summary

   1. The 30* Triangle
The 30* triangle is one of the special right triangles and it requires simplification in order to be able to label the triangle correctly. The first step to the simplification of this triangle is to label it correctly according to the rules of special right triangles. The labeling should be that the hypotenuse is 2x, the leg opposite of the 30* is x, and the longer leg of the triangle is radical 3. To simplify that we must divide the length of each leg and the hypotenuse by 2x because it is the only way our hypotenuse will equal 1, which is our "r." The leg opposite of the 30* should end up being 1/2, our "y," and the longer leg should end up being radical 3/2, the value of our x. When we graph the triangle in the unit circle we draw the coordinate plane and this triangle should be in the first quadrant. The origin of the triangle is (0,0) , going across the horizontal line of the graph we should have (radical 3,2), and when we go upwards from there we get (radical 3/2, 1/2). 

2. The 45* Triangle 
The 45* triangle follows the same rules as the 30 60 90 triangle when it comes to the labeling of the triangle. To simplify the 45 45 90 triangle you must divide by x, except for the two other legs, those must be divided by the reciprocal of 1/x so the x can cancel out and leave you with (radical 2/2) for both legs of the triangle. When graphing it you must use the coordinate plane and it should land in quadrant one, once again, and the origin of the 45 45 90 triangle is (0,0), when you move over on the horizontal line of the graph you should end up with (radical 2/2, radical 2/2), and you should end up with the same when moving up on the graph. 

3. The 60* Triangle 
The same rules as the 30* triangle applies to this one and the only difference is the value change. In this triangle the horizontal value is x and the longer leg is radical 3/2. The labeling still remains the same, the hypotenuse is still 1, the angle that is opposite of the 60* is still y and the horizontal line is 1/2x. When plotting the graph the origin is (0,0), the horizontal points should be (1/2.0) and when you reach the top point, it is (1/2, radical 3/2). 

4. How does this activity help derive the unit circle?
This activity helped me learn that the unit circle is just a bunch of triangles and that they all make up the points in the unit circle, depending on which quadrant they lay on. The quadrant they lay on can determine whether both the x and y are going to be positive, only in quadrant one, in quadrant two only the x is negative, in quadrant 3 both the x and y are negative, and in quadrant 4 the y value is negative. 

5. How do the values change if you draw the triangles in quadrants II, III, and IV? 
The values change depending on the quadrants. If the triangle is in quadrant II then only the x will be negative, if the triangle is in quadrant III then both the x and y will be negative, and if the triangle is in quadrant IV then only the y is going to be negative. 

Inquiry Activity Reflection

The coolest thing I learned from this activity was that all the points on the unit circle are repeated in each quadrant but with different values, like negative and positive. 

This activity will help me in this unit because it helped me to understand where everything in the unit circle comes from and how to differentiate everything that's on the unit circle. With all this information I will be able to fill out the unit circle to help me on the test. 

Something I never realized before about special right triangles and the unit circle is that they are both related to each other. I didn't know that the points in the unit circle came from the special right triangles. 

SP #6: Unit K Concept 10: Writing A Repeating Decimal As A Fraction

This problem goes over how to write a repeating decimal as a fraction without a calculator. The viewer should pay special attention to how the decimals being added has zeros, there are zeros because of the places of the fraction that is going over. The viewer should also pay attention that it is an infinite series and there is an infinity sign when written in summation notation. Be careful that you don't use a calculator on this problem, especially for solving for the ratio.

Fibonacci Haiku: Doge

Doge

Wow

Such math

Doge is everything 

Doge is life and love 
What would life be without math and doge?

http://weknowmemes.com/wp-content/uploads/2013/11/wow-such-math-ti-doge-meme1.jpg

SP 5: Unit J Concept 6: Partial Fraction Decomposition With Repeated Factors

The trickiest part of this problem is counting up the number of factors that there will be in the problem, we must remember that the number of factors will depend on the exponent. The viewer should pay special attention to why there are zeros in the like terms, there are zeros in the like terms because there are no other numbers in the numerator besides 4x^2.

SP 4: Unit J Concept 5 Partial Decomposition With Distinct Factors

The trickiest part of this problem is decomposing the problem back to the original. The viewer should pay special attention to the problem's setup when it is in A, B, and C. The viewer should be careful to not make one little mistake because that little mistake will cause the whole entire problem to be wrong.