A point on the circumference on the unit circle can be associated with an angle, which can be measured in degrees or with an arc length. Note The radius is one. Arc lengths on the unit circle are used to measure angles in a unit called radians. The symbol α, alpha, may either represent a. α0, an angle in degrees; 10 = 1/360 of the degrees in a circle.
b. αradian, an angle in radians, where 1radian is an angle whose arc length equals the radius r.
Q1 1radian = 1r is the same if r = 1 or r = 2. T/F
Relationship between α0 and the arc length αr
00 1800 α0 angle in degrees (rise) I met John Mar in graduate school at Carleton University, Ottawa, Canada. We were taking an applied math course. He
had graduated 10 years earlier as a mechanical engineer and had decided to work in industry instead of pursuing a Ph.D.
He told me it was expensive, and he would not earn money. At that time, he was head of a department at the National Research Council in Ottawa. Seven of his employees had a Ph.D. Obviously he felt uncomfortable, since he only had a mechanical engineering degree. That was the reason he was studying for a master’s degree. After I tell my students this story, I ask them, would John have been in that position if he had obtained a Ph.D.? Personally, I doubt it. He would have worked fewer years. In industry you encounter real problems that require a prompt solution. John Mar learned to solve
The circumference of a circle of radius 1 is 2𝜋.
3600 = (2π)r, the corresponding arc length.
Arc length of a quarter of a circle is 2𝜋/4 = 𝜋/2.
The corresponding arc length of an angle of 300, is 30/360
= 1/12. 300 = (1/12)(2𝜋) = (𝜋/6)r
Who would have thought that going around in circles would be useful in mathematics!
The distance around a circle of radius 1 foot is 6.28 ft.
1. 3600/6.28 = 57.30 is the number of degrees per foot of arc length. T /F 2. If there are 57.30 per foot of arc length, what does (57.3)(6.28) measure?
3. If a circle has radius r, an angle of 1 radian has an arc length r. T/F
them. Unexpectedly John was absent from class for two weeks. He had gone to France to present a scientific paper explaining why a 10-story high balloon, ECO, placed in orbit in 1960 for communication by the Americans, was wobbling.
He explained to me that one side of the metallic surface was overheated by the sun causing it to deform. His presentation was successful.
Length of the arc of a sector α and radius r
Area of a sector of angle α and radius r
Q1 The circumference of a circle of radius is C = _____.
Q2 In the figure, what does π/2 measure?
Q3 Simplify
a) 30/360 = _____ b) 90/360 = _____ c) 150/360 = _____ d) 270/360 = _____
Q4 What do these expressions measure?
a) (30/360)(2π) = _____ b)( 90/360)(2π) = _____ c) (150/360)(2π) = _____
d) (270/360)(2π) = _____ e) (360/360)(2π) = _____
Q5 If the radius is r instead of 1, what do the following expressions measure?
a) (30/360)(2πr) = _____ b)( 90/360)(2πr) = _____ c) (150/360)(2πr) = _____
d) (270/360)(2πr) = _____ e) (360/360)(2πr) = _____
Q6 If the arc length is L, angle α and radius r, the formula L = (α/360)(2πr) can be used to _______.
Q7 In the expression L = (α/360)(2π), the variables are _____ .it is __ function.
Answer Q1 2πr Q2 arc Q3 a) 1/12 b) 1/4 c) 5/12 d) 3/4 Q4 a) 300 arc b) 900 arc Q5 a) 300 arc b) 900 arc Q6 measure L for different α Q7 L, α, linear
πr2 is the area of a circle
Q1 What does (90/360)(πr2).measure?
Q2 What does (180/360)(πr2) measure?
Q3 What does (360/360)(πr2) measure?
Q4 How do you measure the area of a 600 sector of radius 2?
Compare the formulas
Larc = (α/360)(2πr) and Asector = (α/360)(πr2) Q5 How can the formula Larc = (α/360)(2πr) be used?
Q6 How can the formula Asector = (α/360)(πr2) be used?
Answer Q1 Area (1/4) circle Q2 Area (1/2) circle Q3 Area circle Q4 (60/360)(4π) Q5 to find L for known α and r Q6 to find Asector .
45
Lesson 39 A point (x, y) on a circle of radius 1 is ONE unit away from its center. x2 + y2 = 12
Examples where x2 + y2 = 1
1. 00, 12 + 02 = 1
2. 450, (√2/2)2 + (√2/2)2 = 2/4 + 2/4 = 1 Note (√2/2)2 = (√2/2)(√2/2) = √4/4 = 2/4 = 1/2 3. 1500, (−√3/2)2 + (1/2)2 = 3/4 + 1/4 = 1
4. Show that x2 + y2= 1 for a. 300, (√3/2. 1/2) b. 450, (√2/2, √2/2) c. 600, (1/2, √3/2) d. 900, (0, 1) e. 1500, (−√3/2. ½) f. 2100, (−√3/2. −1/2) g. 3300 (√3/2. −1/2) Answer 4. a. 3/4 + 1/4 b. 2/4 + 2/4 d. 0 + 1 e. 3/4 +1/4 f. 3/4 +1/4 g. 3/4 + 1/4
Angle α on the unit circle is associated with point P(x, y), where x = cos α, y = sin α.
Because of their importance, x and y of the unit circle are given special names.
Memorize the run and rise of 00, 300, 450, 600, 900. Use the unit circle and verify these results.
a. cosine 300 = √3/2, sine 300= 1/2, and cos2 300 + sin2 300 = 1 b. cosine 600 = 1/2, sine 600= √3/2 , and cos2 600 + sin2 600 = 1
c. cosine (5π/4)r = −√2/2, sine (5π/4)r = −√2/2, and cos2 (5π/4)r + sin2 (5π/4)r = 1 How to find cos <BAC and sin <BAC of the right triangle ABC where AC ≠ 1
C(3, 4) 5 4
A 3 B
Brain teaser ‒ easier than a previous one
Because water floats in water, one cubic foot of water inside a pool does not rise or sink. One cubic foot of a heavy substance attached to a string is lowered to the bottom of a pool. The force required to pull the object up, must take into consideration the weight of the object and of the water resting on one square foot of surface. T/F
Critical thinking
The earth’s crust is cracked; its pieces, Tectonic Plates, float on a high-temperature fluid substance called magma. The earth’s solid interior is at a temperature of about 5,0000C.
a. If the sun and earth originated in the Big Bang, why are they so different?
a. Is cos <BAC = 3 or cos <BAC = 3/5?
b. cos α is x if r = 1. T/F
AC = 5. If all the sides are divided by 5, <BAC remains the same.
c. cos <BAC = 3/5 and sin <BAC = 4/5. T/F d. Verify 52 = 32 + 42.
Answer a. 3/5 b. T c. T
Cosine and sine are the names of the x and y of a point on the unit circle
x = cos30 y = sin30
b. How and when did part of the interior of the earth become radioactive?
c. In Earth Science textbooks students are taught that initially there was one continent. Supposedly the crust cracked, and newly formed continents drifted away. Since the crust is about 20 miles thick, the only way the earth’s crust could crack is through a huge internal explosion. If the continents drifted away, they are all moving away from their original center of mass. But not all the parts of the Pacific Tectonic plate are moving in the same direction. Some scientists have made significant contributions to science, and some have hindered the progress of science.
Area of a sector
The formula (1/2)sr is similar to the formula of the area of a triangle, (1/2)(base)(height) 5. For α = 300 and r = 2, the point on the circumference is (√3, 1) = 2(√3/2, 1/2).
cos 300 equals a. √3/2 b. √3 c. neither.
6. a. (√3)2 + 12 = ____ b. (√2)2 + (√2)2 = ___
7. s = rαr, find s if a. αr = 2π b. αr = π/2.
Answer 5. √3/2 6. a. 4 b. 4 7. a. 2πr b. πr/2
If math can’t evolve, can the laws of nature evolve?
Only one person, a math graduate from Waterloo University, told me that math appeared through evolution. After telling him logic cannot evolve, he replied: “yes, two plus two is always four”. Several people suggested math was invented. But wait, gravity is math, whoever invented gravity must have invented math. You are right. The majority of people think math was discovered.
I frequently ask, “Could our body be the result of evolution?”
A family doctor said the chemical composition of a human and a monkey are very similar. Yet, it is the pig whose digestive system most resembles that of a human. A PhD. in chemistry from MIT told me the chemical compounds in the stomach of a monkey could evolve into those of a human without violating the laws of chemistry. I was very puzzled, but persevered to follow the logic.
If many of the laws in physics and chemistry can be expressed in mathematical equations, and mathematics cannot evolve, neither can the laws of physics or chemistry. Evolution must be consistent with laws of nature. It cannot contradict itself.
The final question is this: “Can nature improve itself without the interference of an external agent?”
When a ball is dropped from a height H, it can only bounce back to its original height under ideal conditions because the energy lost cannot be recovered, this according to the law of entropy. However, it can bounce back to a height greater than H if an external agent provides additional energy. At best, nature can continue as it is, but never improve itself.
Finally, I wonder, is nature intelligent? What do you think?
When gambling, who wins?
In gambling, is money earned honestly? Those who gamble agree to give part of their money to the winner. Gambling can be so addictive, that even if the person were sure of losing, it is likely the person might continue gambling. And if you hit the jackpot, you may end up with a nightmare. Suppose you have been making $60,000 annually, and suddenly you are given millions of dollars. Now you have lots of new but temporary friends. Who is going to help you administer the money?
Now your life will likely become money centered. Addiction is not beneficial, whether it be gambling, smoking,
pornography or internet. Instead of gambling the person could have done constructive things in life. Shouldn’t we first ask, is this the best I can do? Why give the control of your life to an addiction?
The area of the circle = πr2. (Area of the sector) / πr2 = αr/ 2π.
From this proportion
Area of the sector = (πr2)(αr/2π) = (1/2)(r2 αr) = (1/2)(r2)(s/r) = (1/2)sr
Note the formula is like that of a triangle
4. Find the area of a sector if the radius is r and a. s = πr b. αr = π c. s = 2πr
Answer a. (1/2)πr2 b. (1/2)(r𝜋)r = 1/2)πr2 c. πr2
47