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Chapter 4 Trigonometric Functions Background •The term trigonometry is derived from ancient Greek roots meaning “three side measure”. •It is the study of ratios within right triangles. •Trigonometry came about in the ancient world when the transition was made from a flat earth to a world of circles and spheres. •Today it is used to understand everything from electrical current to modern telecommunications. •Any type of periodic (recurring) behavior can be modeled using trigonometric functions. Section 4.1 Angles and Their Measure Warm-up Warm-up Angle Vertex 90° Standard y Position 180° 0° O Initial side x 270° Degree •A degree, represented by the symbol °, is a unit of angular measure equal to 1/180th of a straight angle. •In the DMS(degree-minute-second) system of angular measure, each degree is subdivided into 60 minutes (denoted ´) and each minute is subdivided into 60 seconds (denoted´´) Example 1: Working with DMS Measure 1. Convert 37.425° to DMS. 2. Convert 42°24´36´´ to degrees. Radians A radian is the measure of a central angle θ that intercepts an arc “s” equal in length to the radius “r” of the circle. s θ , y r s=r r where θ is θ r measured in radians. x Radians are the angle measures used in calculus. 360° = 2π radians 2π 360 1 1 radian 360 2π π 180 1 1 radian 180 π Formula to go from Formula to go from degrees to radians. radians to degrees. π y rad 2 π rad x 0 rad 3π rad 2 Example 2: Working with Radian Measure 1. How many radians are in 60 degrees? 2. How many radians are in 150 degrees? Example 2: Working with Radian Measure 3. How many degrees are in π/4 radians? 4. How many degrees are in 3π/5 radians? Arc Length For a circle of radius “r”, a central angle θ intercepts an arc of length “s” given by s = rθ where θ is measured in radians. Example 3: Finding arc length 1. A circle has a radius of 27 inches. Find the length of the arc intercepted by a central angle of 160°. Give the answer in terms of π and rounded to the nearest hundredth. 1. Change 160° into radians. π 8π 160 rad 180 9 2. Find the arc length. 8π s 27 24π inches 75.40inches 9 Example 3: Finding arc length 2. Find the perimeter of a 30° slice of a large 8in. radius pizza Example 3: Finding arc length 3. A 100-degree arc of a circle has a length of 7 cm. To the nearest centimeter, what is the radius of the circle? Example 3: Finding arc length 4. The concentric circles on an archery target are 6 inches apart. The inner circle (red) has perimeter of 37.7 inches. What is the perimeter of the next largest (yellow) circle? Note: radius = 1 radian Example 4 John’s truck has wheels 36 inches in diameter. If the wheels are rotating at 630 rpm (revolutions per minute), find the truck’s speed in miles per hour.

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posted: | 9/1/2012 |

language: | English |

pages: | 24 |

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