SP7: Unit Q Concept 2 IDENTITIES

This WPP13-14 was made in collaboration with Louis Troung.  Please visit the other awesome posts on their blog by going here.

 

 

WE CAN FIND THE SAME THING USING SOHCAHTOAH

I/D#3: Unit Q Concept 1: Pythagorean Identities

INQUIRY ACTIVITY SUMMARY

     Pythagorean identities come from the Unit Circle and the Pythagorean Theorem. An Identity is an equation that is true no matter what values are chosen. The Pythagorean Theorem kind of looks like an  identity because even if you have 2 sides you can always find the 3rd using the theorem. The Pythagorean theorem is a^2+b^2=c^2 and the Pythagorean identity is cos^2(theta) + sin^2(theta) = 1. but on a unit circle it is x+y=1  we are going to substitute them so a is going to be x,  y is b and 1 = r so 1 is c as you can see. r=1 because r is the radius of the circle and the radius of a unit circle is always going to be 1.

\what you can notice about how cosine =x and sine= y that it really does relate because the equation is cos^2(theta) + sin^2(theta) = 1 and you can see cos = x in the equation and sin=y and 1 =1. just like the Pythagorean theorem they are squares and like the unit circle they are x and y. and this is one of the identities.  

 2nd identity

 

3rd identity
1. the connection I have seen between Units N,O,P, and Q are that we can find angles on the unit circle and the sides and angles inside triangles that are missing using law of sines and cosines
2) If I had to describe trigonometry in THREE words, they will be hard, understandable, and progressive.

Wpp13-14:concepts6-7 applications law of sines and cosine

WPP 13 & 14
 This WPP13-14 was made in collaboration with Louis Troung.  Please visit the other awesome posts on their blog by going here.
 Stephanie decides to go on a vacation with Louis after all her soccer tournaments were over.they end up going to a beachside house. there are two boats but they are both different colors one is blue and the other is red. Stephanies favorite color is blue just like louises so they are going to race for it. Stephanie is 10 feet away from Louis . Stephanie knows she ahs to turn
S30E and Louis need to turn N32E to get to the boat. which one of them is closer to the boat?

this is the visual we get from the word problem

 once we start using the hints we got for the angles we can now see that this triangle is an angle side angle and we can figure this out by using law of sines

as you can see in this picture we are going to figure out the missing angle that is left there bcause it will be the one pair we need to use law of sines we are going to subtract 180 from 32 and 30 because all triangle angles add up to 180 so that's how we are going to find out the missing angle after that you will just stat using the law of sines as you see in the picture leaving us with Louis being closer thus him getting to the boat faster.


On a nice day Stephanie and louis decide to go boating on our boats ! Stephanie and louis are 10 miles apart . Stephanie and louis know that louis has a bearing of 043 to Stephanie. The light house is 40 miles away from louis to the east I him .  Stephanie wants to go to the light house. How far is she from the light house ?

BQ#1 Unit P: Concepts 1 and 4 law of sinces and oblique trangles

Concept 1: Law Of Sine

We need to use law of sines when we need to find sides of a triangle when it isn't a special right triangle. But we can only use it if the trangle is AAS or ASA


We can derive the law of sines with this video on top .

Concept 4 Area Of Oblique Triangles.

we can derive this from the area of a triangle. we have information about the triangle but not about the height so we are missing that part for the equation so what we can do is use the area of oblique triangles. the equation is A=1/2 *(2 sides)*sine of the angle that is given. we can use this because as you can see on the picture above we used sine of c anf that will be sinc= h/a . after that we will multiply a to both sides and get H=asinec  and since we needed h we plug that in into the area of the triangle and that's how we get our equation. we can work out a problem on the pictures on bottom

WPP#12 Unit O Concept 10 finding elevation and depression

Stephanie is playing a soccer game and when she kicks the ball she accidentally hits a girl in the face. this girl is 5.5 feet tall that's from her foot to her head the angle of elevation the ball went was 24.5. knowing this how far did the ball travel .

Stephanie has won a toy helicopter at the soccer game and it was 15 feet up in the air. she wanted too ply with it by making it land like an airplane, the toy helicopter is 20 feet away from where she wants to land it what is the angle of depression

ID#2 Unit O Concept 7-8

INQUARY AVTIVITY SUMMARY

 

 

So basically in this paper we are going to derive an equilateral square and triangle. With these we are going to get a 45-45-90 triangle and the 30-60-90 triangle. when we cut the in half we will get these special right triangles and for us to solve the missing side we are logically going to use the Pythagorean  theorem.

45-45-90 Triangle

In this picture you can see what we are starting off with, it is an equilateral square.

Here we can see that we already cut it in have so since all sides are 90 when you cut it in half your other angles are going to be 45 degrees.

Here I am doing the work for the use of the Pythagorean theorem. we will use a^2+ b^2 = c^2 since we already have both sides it is obviously one and since we have to take the radical to get c alone it will end up being radical 2 for out hypotenuse answer.

Here we can see that the both sides will always be the same number so we can go ahead and substitute it got "n" and then for the hypotenuse we will always have to multiply with radical 2 so for example we have one side of the 45 degree angle 5 the other side must me 5 and the hypotenuse 5 radical 2.

  30-60-90- Triangle

Here we can see how we have our equilateral triangle

now here we have already but the triangle in half and we have all the degrees there too. all sides on an equilateral triangle are 60 degrees and when you cut it in half we will have 30 and 30 degrees on the sides.

So when we have to do the Pythagorean theorem we can see that the bottom will have to split into 2 so one divided by 2 is one half. when we have that we can plug everything in. we edn up getting radical 3 over 2 for side b.

 

Here we can see that we really don't want to use the radial 3 over 2 because it will make it difficult so we are going to multiply everything by 2 to make it easier when we do we are going to add an n to represent any number so when we have that we and up getting one for side a radial 3 for side b and 2 for the hypotenuse but got this triangle there are different degrees so the 60 side as you can see has the radical 3 so it will always have to do with that. when we have n equal to 6 side b will be 6 radial 3 and the hypotenuse will be 12. when we do the problem backward for example and b equals 8 we will have to go back to the a side by dividing by radial 3 which will give us 8 rad 3 over 3 and the hypotenuse 16 rad 3 over 3 .

INQUIRY ACTIVITY REFLECTION

1. Something I never noticed before about special right triangles was that they all had the same ration between each other and how we found that out.
2. Being able to derive these patters myself aids in my learning because i know how i got there and how those triangle were made