During our morning class, Mr. P first went over the Special Angles and Multiples and Fill in The Unit Circle sheets that he assigned for homework the day before. In case you missed it, here they are:
(I would like to apologize in advance for the tiny notes/scribbles on the sides of the sheets. Please ignore them!)
As you can see, the sinθ and cosθ values for the angles and their multiples repeat after crossing the x or y axis (they just differ in sign). This is due to their location on a coordinate plane (quadrant).
30°-90°= Quadrant 1
120°-180°= Quadrant 2
210°-270°= Quadrant 3
300°-360°= Quadrant 4
This is all in reference to the CAST RULE.
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Next, we looked at the Unit Circle sheet. It has all of the degrees, multiples, exact radians, and the coordinates for sin and cos.
After that, the class proceeded into solving for the exact coordinates of P(θ).
To solve these questions, you must first understand the concept of reference angles, special triangles, co terminal angles, SOH CAH TOA, and the CAST rule. If you are unsure about these, I strongly suggest checking out my friend, Jasiel Rosario's post that explains the basics of Special Right Triangles.
a) P(13π/2)=(x, y)
Just by looking at the given value, we can already see that our reference angle is π/2.
If π =180°, that must mean that π/2 = 90°. Looking back at the Unit Circle sheet, π/2 falls on the y-axis with the coordinates (0,1). Therefore, P(13π/2) also has the same coordinates. If you are confused as to why this is, I suggest counting counter-clockwise by 90°s (1/4) around the circle for 13 times. You will end up on the y-axis.
Answer:
P(13π/2)=(x, y)
P(13π/2)=(0,1)
b) P(-9π/4)=(x,y)
Our reference angle this time will be π/4. We can use this information to figure out what type of special triangle we are dealing with.
If π =180°, π/4= 45°. Our special triangle is:
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| Opp=1, Adj=1, Hyp=√2 |
We learned during our previous classes that (x, y)=(cosθ, sinθ). With this in mind, we can start solving for the coordinates!
First, we need to find cos and sin. According to SOH CAH TOA, cos = adjacent/hypotenuse and sin =opposite/hypotenuse.
In this case, cos = 1/√2 and sin = 1/√2.
Then you plug these into (cosθ, sinθ)! You will end up with (1/√2, 1/√2).
The final step is to apply the CAST rule. This will help you determine the signs of your coordinates. If you were to count clockwise (because negative) around the circle by 45°s (1/8) for 9 times, you will find yourself in Quadrant 4. Quadrant 4 is the "C" section, meaning everything else but cos must be negative. After applying this to the coordinates we obtained earlier, the answer should be (1/√2, -1/√2).
Answer:
P(-9π/4)=(x,y)
P(-9π/4)=(1/√2, -1/√2)
You will need to follow the same steps in order to solve c) and d).
During our afternoon class, we learned how to solve for the measurement of central angle θ when given an interval.
This is basically like the previous questions we solved but reversed. Remember, the given interval is 0≤θ≤2π.
a) P(θ)=(0,1)
Based on the question we solved earlier, we know that the coordinates for π/2 is (0,1) because it lies on the y-axis. Therefore, the the measurement of θ should be π/2.
Answer:
P(θ)=(0,1)
P(θ)=π/2
b) P(θ)=(-1/√2, -√3/2)
The given cos and sin value are negative. According to the CAST rule, the only quadrant where cos and sin are both negative is quadrant 3.
Because the cos value is (-1/√2) and the sin value is (-√3/2), our special triangle must be:
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| Opp=√3, Adj=1, Hyp=2 |
We can now start labeling our special triangle.
Opp= -√3, adj= -1, hyp= 2 (hypotenuse will always be positive). The degree is 60.
Now that we know what the degree is, we can find the reference angle. The reference angle for this question is π/3. All we have to do now is to count counter-clockwise by 60°s (1/6) around the circle until we reach our triangle. It should take us 4 times. Therefore, our θ should be 4π/3.
Answer:
P(θ)=(-1/√2, -√3/2)
θ= 4π/3
You will need to follow these steps in order to solve c) and d).
Mr. P then proceeded into teaching us how to find the coordinates of P(θ) when it is rotated.
There are certain patterns we can use when determining the coordinates of points on the unit circle!
Pattern #1: When asked to rotate the original θ to a half rotation (π), P(θ ± π) will result in the same x and y values but their signs will depend on which quadrant they are on.
Example 2:
P(θ ± π)=(3/5, -4/5), find exact coordinates of P(θ+π) and P(θ-π).
Based on the coordinates given:
cosθ=adj/hyp
cosθ=3/5
sinθ=opp/hyp
sinθ=-4/5
If we plot this point on our circle, it would be on quadrant 4. This is because our cos value is positive and sin value is negative (CAST rule).
Now if we rotated it by half or π(direction does not matter in this case because half rotations in either direction will result in the same point), we will end up in quadrant 2.
We then need to apply the CAST rule. Because we are now in quadrant 2, our cos must be negative and our sin must be positive. The point will then have the coordinates (-3/5, 4/5).
Answer:
P(θ ± π)=(-3/5, 4/5)
Pattern #2: When asked to rotate original θ to quarter rotation, P(θ ± π/2) will result in switched x and y values. Just like the previous pattern, the signs depend on the quadrant.
Example 2:
P(θ)=(3/5, -4/5), find the exact coordinates of P(θ+π/2) and P(θ-π/2).
Based on the given coordinates:
cosθ=adj/hyp
cosθ=3/5
sinθ=opp/hyp
sinθ=-4/5
If we plot this point on our circle, it would be on quadrant 4.
Now if we rotate it clockwise by a quarter(π/2), we will end up in quadrant 3. If we rotate it counter-clockwise, we will end up in quadrant 1 (direction in this case matters because both ways will result in different points).
The point on quadrant 3 will have the coordinates (-4/5, -3/5). The values are switched and both must be negative (CAST rule).
The point on quadrant 1 will have the coordinates (4/5, 3/5). The values are switched and both must be positive (CAST rule).
Answer:
P(θ-π/2)=(-4/5, -3/5)
P(θ+π/2)=(4/5, 3/5)
Well, that's about it! I hope my explanations were clear and helpful :) If you are still having trouble understanding these lessons, here's a video that might help you: http://www.youtube.com/watch?v=hzucE6S5SXM. Have a great weekend and see y'all on Monday!!!
<3 Pat
P.S. I was browsing through the Internet yesterday (yes, before I fell asleep) and BAM! I saw this:
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| As I am writing this, my mom is staring at me with a WTH face because I'm laughing like a maniac. Thank you Clay Aiken. |
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