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LECTURE BREAK 01 – Angle Measure in Degrees

LECTURE BREAK 01 – Angle Measure in Degrees Working in groups of two or three, discuss and answer the following questions : 1. Sketch a central angle of approximately 100 degrees. 2. Sketch a central angle of approximately 300 degrees.

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LECTURE BREAK 01 – Angle Measure in Degrees

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  1. LECTURE BREAK 01 – Angle Measure in Degrees • Working in groups of two or three, discuss and answer the following questions: • 1. Sketch a central angle of approximately 100 degrees. • 2. Sketch a central angle of approximately 300 degrees. • 3. At the given times, what is the angle between the minute hand and the hour hand on a clock face? • a) 3:00 d) 4:00 • b) 6:00 e) 7:00 • c) 10:00 f) 11:00

  2. WHY 360˚ ? • The ancient astronomers noticed that the stars seem to rotate around the celestial pole about 1/360th of a full rotation every day. • 360 is ALMOST the number of days in a year. Some ancient calendars used 360 days. • 360 is a multiple of 60, the base of the Sumerian number system (3500 – 3200 BC). Sumerians were the first to use 360 degrees, with 60 arc seconds each, to represent the circle. They also created the concept of “place value”. • 60 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30, making it very easy to work with by hand. The Sumerian base-60 system (termed ‘sexagesimal’) was also able to express the large numbers used in the commercial and agricultural calculations of the time. The Sumerians lived in the Fertile Crescent region – present day Iraq. • Babylonian astronomers inherited the Sumerian mathematical system and used it to make advanced celestial calculations for the first time, successfully predicting events such as eclipses. • The Greek Hipparchos, 190 BC – 120 BC, a famous astronomer and mathematician, and considered the father of trigonometry, adopted the 360 degree circle model for his calculations.

  3. WHY RADIANS ? • The ratio of arc length to radius was used from the early 1700’s because it was more efficient in certain calculations and formulas. For example, Swiss mathematician Leonard Euler used radians in his very famous formula relating the exponential function to the trigonometric functions and the complex number system (1748): (in this case must be in radians) • 360 is an arbitrary number (similar to Fahrenheit temperature) that fits poorly with the rest of the International System of Units (SI units). Radians are the established SI unit for angle measure. • Radian measure is dimensionless (unitless), which makes several geometric formulas much easier to state if is in radians. For example:arc length = sector area = • In calculus, radian measure is the standard and the basis for many advanced calculus concepts.

  4. LECTURE BREAK 02 – Radian Measure • Working in groups of two or three, discuss the following questions: • 1. State the formula for the radian measure of using , radius , and arc length . • 2. Fill in the table below: • 3. Discuss a way to convert from degrees to radians and radians to degrees for any angle. Try to make the following conversions: • a) to radians • b) the angle in radians between the hour hand and the minute hand at 10:00 • c) 3 radians to degrees

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