Download 6.1 Radian and Degree Measure

Foundations of PreCalculus Chapter 6: Graphs of Trigonometric Functions Section 6-­‐1: Angles and Radian Measure SWBAT: change from radian measure to degree measure and vice versa. Find the length of an arc given the measure of the central angle. Find the area of a sector. Common Core: •
G-­‐C.B.5 Recall our Unit Circle from Chapter 5. On the inside of our unit circle we had two types of angle measurements. Ø Degrees Ø Radians (the angle measurement in terms of 𝜋). v The definition of radian is based on the concept of the Unit Circle in which the radius is one unit of length. v A point 𝑃(𝑥, 𝑦) is on the unit circle if and only if its distance from the origin is 1. Thus, for each point 𝑃(𝑥, 𝑦) on the unit circle, the distance from the origin is represented by the following equation: Therefore, the _________________________ measure of an angle in standard position is defined as the length of the corresponding arc on the unit circle. Ø A full revolution around the unit circle corresponds to an angle measure of ____________________. Why? Foundations of PreCalculus Chapter 6: Graphs of Trigonometric Functions v This idea that 1 full revolution around the unit circle (𝟑𝟔𝟎°) is equivalent to a radian measure of 𝟐𝝅 can be used to help us convert back and forth from degrees to radians and vice versa. Converting between Degrees/Radians: We know that 360° = 2𝜋, so… Ø Degrees to Radians multiply by ____________________. Ø Radians to Degrees multiply by ____________________. Example 1: a) Change 330° to radian measure. b) Change 115° to radian measure. d) Change −
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c) Change ! radians to degrees. !!
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radians to degrees. Example 2: a) Evaluate cos
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b) Evaluate tan
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d) Evaluate sec −
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c) Evaluate csc
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Foundations of PreCalculus Chapter 6: Graphs of Trigonometric Functions Ø Radian measure can be used to find the length of a __________________________________________. Ø A circular arc is a part of a circle. The arc is often defined by the ________________________________ that intercepts it. A central angle of a circle is an angle whose vertex lies at the center of the circle. Arc Length: The length of any circular arc s is equal to the product of the measure of the radius of the circle and the radian measure of the central angle. In Degrees In Radians Example 3: a) Given a central angle of 128°, find the arc length if the circle has a radius of 5 centimeters. Round to the nearest tenth. b) Given a central angle of 3.2 radians, find the arc length if the circle has a radius of 12 centimeters. Round to the nearest tenth. Foundations of PreCalculus Chapter 6: Graphs of Trigonometric Functions Example 4: Winnipeg, Manitoba, Canada, and Dallas, Texas, lie along the 97°W longitude line. The latitude of Winnipeg is 50°N, and the latitude of Dallas is 33°N. The radius of Earth is about 3960 miles. Find the approximate distance between the two cities. Ø A _______________________ of a circle is a region bounded by a central angle and the intercepted arc. Recall that the area of a circle is…𝐴 = 𝜋𝑟 ! To find the area of a sector, we are finding a fraction of the circle based off of the length of the arc that we are given. Foundations of PreCalculus Chapter 6: Graphs of Trigonometric Functions Example 5: !!
a) Find the area of a sector if the central angle measures ! radians and the radius of the circle is 16 centimeters. Round to the nearest tenth. b) Find the area of a sector is the central angle measures 82° and the radius is 11 feet. Round to the nearest tenth. Homework: Handout 6.1