SP7: Unit Q Concept 2- Finding Trig Function When Given One Trig Function and Quadrant

This SP7 was made in collaboration with Angela Luong.  Please visit the other awesome posts on their blog by going going here.

I/D #3

Inquiry Activity Summary
1.Today we will be looking at the devriation of the "identity". "Identity" is a proven facts and formula that is always proven true. For example, the pythagorean theoream of a^2+b^2=c^2 is an identity.

To begin with, the pythagorean theorem of a triangle using x, y, and r, is x^2+Y^2=r^2. To make it equal to 1, we would have to divide everything by r^2. Since the ratio for cosine on the  unit circle is x/r and the ratio for sine is y/r, we can say that cos^2theata+sine^2theata=1. Therefore, we can conclude that sin^2x+cos^2x=1 is a pythagorean identity because it is proven to be exactly the same as x/r^2+y/r^2=1. The replacement of x and y with cos and sin proves that it is true.

Another way that is proven true is to plug in the "magic 3" ordered pairs from the unit circle. When we plug in the pairs, we know that it is equal to 1. Therefore, it is an identity.

2. To derive the identity from sin^2+cos^2x=1, first divide everything by cos^2. Sin/tan is equal to tan. Because it is squared with an x, we leave that the same. Cos^2x/cos^2x is equal to 1. 1/cos^2x is equal to sec^2x

To derive it from cosecant and cotangent, you set up the same. Sin2^x+cos^2x=1. To begin with, you divide everything by sin^2x. Sin^2x/sin^2x=1. Cos^2x/sinx^2 is equal to cot^2x. 1/sin^2x is equal to csc^2x.

Inquiry Activity Reflection:

The connections that I see between Units N, O, P, and Q so far are unit circle has relation between these identities. These identities is basically derived from the pythagorean theorem. The pythagorean theorem can also derived from these triq functions.

If I had to describe trigonometry in THREE words, they would be identity, magic ordered pairs, and pythagorean theorem.

WPP #13 & 14: Unit P Concept 6 & 7

This WPP #13-14 was made in collaboration with Angela Luong. Please visit the other awesome posts on their blog by going here

Law of Sines: 

Amber is walking up the street 13 feet towards a store called "Frosty" due North of her car. Her friend sends her a text to get her a cupcake from the store "Cakes-galore!", the shop is located at a bearing of 046*; the "Frosty" store is located at a bearing of 135*. How far is each store from Amber? 

Law of Cosines: 

Amber is walking down to her new job at Clothing Express located 12 feet due North. She hears a cry of "Thief!" and turns her head 32* to see a man carrying a bag of money running 17 feet due East. Amber tries to run after him. How far is the man from Amber? 

BQ: Law of Sin and Area Formula

Law of Sin

To being with, we will be looking at the law of sin and how it is derived to get the actual formula. The law of sin begins with  a triangle. When it is cut through, it give us a hypotenuse.

In trigonometry, we know that sin is opposite over hypotenuse. Therefore, we know that the sin of angle A is equal to h over c. You multiply both sides by c to get h alone 

Next we must find the law of sin for angle C. Using soh-cah-toa the sin of angle C is h over a. We must isolate the h by multiplying a to get aSinC.

Finally we can use the transitive property to derived to find the real sin formula. You take the answers to both angle and set it equal to each other. Divide by ca to both side to get SinA over side a and SinC equals to side c. There, you get the toe ratio of the law of sin 

You can only use the law of sin if you have AAS(angle, angle, side), ASA(angle, side, angle), and SSA(side, side, angle).

The case when you cannot use the law of sin is when you don't know the opposite side or angle. For instance like in this picture it is given the side but the angle is missing. In addition to, the angle is also given but the opposite side is missing. 

4.Area Formula

The area of an oblique triangle is derived with a combination of the law of sin and a regular area of a triangle. You would use the same formula of 1/2 times based times height with the substitution of h. To find h, you use the law of sin, h is equal to aSinC. There, you have the devriation of the oblique triangle.  Also for it to be an a oblique triangle, all sided must be different.

There is a several different form of the area depending in the angles you are looking for.

Reference:

Math Warehouse

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

Christine and Jessica wanted to promote their newly open clothing shop. In order for potential customers to see the shop they decided to put a banner on the top of the clothing shop. Jessica is standing away from the building to check on the sign. The angle of elevation is 25.25 degress from jessica to the top where Christine is at. IF the base of the building is 350 ft from Jessic how high is the building?

After finishing up with the sign Christine wanted to use the latter down. She estimated the angle of depression of 30 degress at where she is now to where Christine is at. She is 200 ft higher than the base of the building. How long is her way down the building?

I/D#2: Unit O Concept 7-8: Deriving the pattern of SRT

Inquiry Activity Summary:

45,45,90 degrees SRT
To begin, we need to label each side equals to 1. Next, the equallateral needs to be cut in half in order to find the 45,45,90 degrees triangle. 

Since the given information is only 2 sides of the triangles, we need to find the missing sides. You can find the missing side by using the pythagorean theorem of a^2+b^2=c^2. The legs of the triangle will always A and B while the hypothenus is C. When it is plugged in, it should be (1)^2+(1)^2=C^2. After solving, you should get C= radical 2.

For it to be a SRT, N should be added to the pattern because the value can be change anytime. The final pattern of the 45,45,90 should be 1n,1n, and N radical 2. 

30,60,90 SRT

Now we are going to look at another special right triangle. It is the 30,60,90 degrees. To begin, we label each side equalss to 1. Then we cut the triangle in half.

Next, the 1 on the short leg becomes 1/2 because the triangle has been cut half. The hypotenus will stay as a value of 1. The pythagorean theoream needs to be use in order to find the other missing side of the triangle

The pythagorean theorem is a^2+b^2=c^2 After solving, you should get C=radical 3/2

Since the value of each side does not follow the pattern, we must find the pattern by multiplying everything by 2. 

Finally, we should get the pattern of 1,2,and radical 3. However, the n needs to be there because the value of these can be change depending on the triangle.

Inquiry Activity Reflection:

The coolest thing I learned from this activity was that we can pattern came from a triangle that has all side equals to 1.

The coolest thing I learned from this activity was I can derive the formula if I were to forget it during the test or in the future.