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May 28 Math 1113 sec 002 Summer 2014 Arclength Formula (sec. 6.1) Given a circle of radius r , the length s of the arc subtended by the (positive) central angle θ (in radians) is given by s = r θ. The area of the resulting sector is A = 12 r 2 θ. () May 28, 2014 1 / 50 Motion on a Circle: Angular & Linear Speed Definition: (angular speed) If an object moves along the arc of a circle through a central angle θ in the time t, the angular speed is denoted by ω (lower case omega) and is defined by θ ω= . t Definition: (linear speed) If the circle has radius r , then the distance traveled is the arclength s = r θ. The linear speed is denoted by ν (lower case nu) and is defined by ν= () s rθ = = r ω. t t May 28, 2014 2 / 50 Example () May 28, 2014 3 / 50 () May 28, 2014 4 / 50 () May 28, 2014 5 / 50 Example A child is sitting on the edge of a merry-go-round that makes three full revolutions per minute. If the radius of the merry-go-round is 6 ft, what is her linear speed. () May 28, 2014 6 / 50 () May 28, 2014 7 / 50 Caveat! Remember that the formulas for arclength, sector area, angular speed, & linear speed are for an angle in radians. An angle in degrees must be converted to radians before applying any of these formulas. () May 28, 2014 8 / 50 Section 6.2: Trigonometric Functions of Acute Angles In this section, we are going to define six new functions called trigonometric functions. We begin with an acute angle θ in a right triangle with the sides labeled: () May 28, 2014 9 / 50 Sine, Cosine, and Tangent For the acute angle θ, we define the three numbers as follows sin θ = cos θ = tan θ = opp , hyp read as ”sine theta” adj , hyp read as ”cosine theta” opp , adj read as ”tangent theta” Note that these are numbers, ratios of side lengths, and have no units. It may be convenient to enclose the argument of a trig function in parentheses. That is, sin θ = sin(θ). () May 28, 2014 10 / 50 Cosecant, Secant, and Cotangent The remaining three trigonometric functions are the reciprocals of the first three csc θ = 1 hyp = , opp sin θ sec θ = cot θ = () hyp 1 = , adj cos θ adj 1 = , opp tan θ read as ”cosecant theta” read as ”secant theta” read as ”cotangent theta” May 28, 2014 11 / 50 Example Determine the six trigonometric values of the acute angle θ. () May 28, 2014 12 / 50 Example Determine the six trigonometric values of the acute angle θ. () May 28, 2014 13 / 50 () May 28, 2014 14 / 50 Some Key Trigonometric Values Use the triangles to determine the six trigonometric values of the angles π4 , π3 and π6 . () May 28, 2014 15 / 50 () May 28, 2014 16 / 50 () May 28, 2014 17 / 50 Commit To Memory It is to our advantage to remember the following: sin π6 = 12 , cos π6 = tan π6 = √ 3 2 , √1 , 3 √ sin π4 = √1 , 2 sin π3 = cos π4 = √1 , 2 cos π3 = tan π4 = 1, tan π3 = 3 2 1 2 √ 3 We’ll use these to find some other trigonometric values. Still others will require a calculator. () May 28, 2014 18 / 50 Calculator Figure: Remember to check the mode for radians or degrees. (TI-84 shown) () May 28, 2014 19 / 50 Using a Calculator Evaluate the following using a calculator. Round answers to three decimal places. sin π = 8 sec 78.3◦ = tan(0.2) = () May 28, 2014 20 / 50 Application Example Before cutting down a dead tree, you wish to determine its height. From a horizontal distance of 40 ft, you measure the angle of elevation from the ground to the top of the tree to be 61◦ . Determine the tree height to the nearest 100th of a foot. () May 28, 2014 21 / 50 () May 28, 2014 22 / 50 Application Example The ramp of a moving truck touches the ground 14 feet from the end of the truck. If the ramp makes an angle of 28.5◦ with the ground, what is the length of the ramp? () May 28, 2014 23 / 50 () May 28, 2014 24 / 50