SECTION 7.5 Sum and Difference Formulas 523 (c) Find the exact area of the circle inscribed in a regular dodecagon with side = a 5 cm. (d) Find the exact area of the region between the circle and the regular dodecagon. 107. Geometry: Angle between Two Lines Let L1 and L2 denote two nonvertical intersecting lines, and let θ denote the acute angle between L1 and L2 (see the figure). Show that m m mm tan 1 2 1 1 2 θ = − + where m1 and m2 are the slopes of L1 and L ,2 respectively. [Hint: Use the facts that θ = m tan 1 1 and θ = m tan . 2 2 ] x y u2 u L1 L2 u1 108. Challenge Problem Show that = − − − e e cot tan . v v 1 1 109. Challenge Problem Show that π + = − − v v sin cos 2 . 1 1 110. Challenge Problem Show that π + = − − v v tan cot 2 . 1 1 111. Challenge Problem Show that π ( ) = − − − v v tan 1 2 tan , 1 1 if > v 0. 112. Challenge Problem If α β γ + + = ° 180 and θ α β γ θ = + + < < ° cot cot cot cot 0 90 show that θ α θ β θ γ θ ( ) ( ) ( ) = − − − sin sin sin sin 3 113. Challenge Problem If α = +x tan 1 and β = −x tan 1, show that α β ( ) − = x 2 cot 2 105. Area of a Dodecagon Part I A regular dodecagon is a polygon with 12 sides of equal length. See the figure. a r (a) The area A of a regular dodecagon is given by the formula π = A r 12 tan 12 , 2 where r is the apothem, which is a line segment from the center of the polygon that is perpendicular to a side. Find the exact area of a regular dodecagon whose apothem is 10 inches. (b) The area A of a regular dodecagon is also given by the formula π = A a3 cot 12 , 2 where a is the length of a side of the polygon. Find the exact area of a regular dodecagon if the length of a side is 15 centimeters. 106. Area of a Dodecagon Part II Refer to Problem 105. The figure shows that the interior angle of a regular dodecagon has measure ° 150 , and the apothem equals the radius of the inscribed circle. a r 1508 (a) Find the exact area of a regular dodecagon with side = a 5 cm. (b) Find the radius of the inscribed circle for the regular dodecagon from part (a). Explaining Concepts 114. Discuss the following derivation: θ π θ π θ π θ π π θ θ θ θ ( ) + = + − = + − = + − = − = − tan 2 tan tan 2 1 tan tan 2 tan tan 2 1 1 tan 2 tan 0 1 0 tan 1 tan cot Can you justify each step? 115. Explain why formula (7) cannot be used to show that π θ θ ( ) − = tan 2 cot Establish this identity by using formulas (3a) and (3b).
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