(9/30/08)Math 10A. Lecture Examples.Section 1.5. Trigonometric functions†Example 1 Find the cosine, sine, and tangent of the angle ψ (psi) in the right triangleof Figure 1.ψ53FIGURE 1Answer: cos ψ =3√34• sin ψ =5√34• tan ψ =53Example 2 The largest ferris wheel of all time was designed by George Ferris and builtfor the 1893 World’s Columbian Exposition in Chicago. It had a 125 footradius and could carry 2160 passengers.(1)Suppose that the ce nter of thewheel was 150 feet above the ground and that it turned counterclockwiseas viewed in the schematic sketch with a uv-plane in Figure 2. The dotrepresents a car on Ferris’ wheel with coordinates (u(θ, v(θ)) at the angle θ(radians). (a) Which of Figures 3 and 4 shows the graph o f u = u(θ) andwhich shows the graph of v = v(θ)? (b) Give formulas for u(θ) and v(θ).u−100 100v (feet)300(feet)θθ100200300π 2πθ100−100π 2πFIGURE 2 FIGURE 3 FIGURE 4Answer: (a) u = u(θ) is in the second dr awing. • v = v(θ) is in the first drawing.(b) u(θ) = 125 cos(θ) • v(θ) = 150 + 125 sin θ†Lecture notes to accompany Section 1.5 of Calculus by Hughes-Hallett et al.(1)“The Fer ris Wheel” by H. Petrovski, American Scientist, May-June, 1993, pp. 216–222.1Math 10A. Lecture Examples. (9/30/08) Section 1.5, p. 2Example 3 Figure 5 shows the graph of a sinusoidal function. Give its formula.x1y5FIGURE 5Answer: y = 5 sin12πxExample 4 Figure 6 shows a vertical pole 10 fee t high and its shadow that is cast on thehorizontal ground by the setting sun. Find a formula for the length s = s(θ)of the shadow as a function of the angle θ between the rays of the sun andthe pole .10 feetShadowSunθGroundsPoleθs (feet)50100150s = 10 tan θ(radians)14π12πFIGURE 6 FIGURE 7Answer: s(θ) = 10 tan θ, 0 ≤ θ <12πExample 5 Figure 7 shows the graph of s = s(θ) from Example 4. Use Figure 6 (a) toexplain why s(0) = 0, (b) to explain why s(θ) is larger for larger θ, and(c) to find the value of s14π.Answer: (a) s(0) = 0 because there is n o shadow when the sun is directly over the pole.(b) s(θ) is larger for larger θ because the sha dow is long er when the sun is lower in the sky.(c) s(14π) = 10 because with the sun at this ang le the right triangle in Figure 6 is isosceles.Section 1.5, p. 3 Math 10A. Le cture Examples. (9/30/08)Example 6 What is the angle γ in the right triangle of Figure 8?γ125FIGURE 8Answer: γ = sin–1512.= 0.42978 radiansExample 7 Ravenna, Italy is 73 kilometers nort h and 76 kilome ters e ast of Florence(Figure 9). (a) How far is Ravenna from Florence? (b) What is thedirection from Florence toward Ravenna?FIGURE 9Answer: (a) [Distance] =√732+ 762.= 105 kilometers(b) The direction from Florence to Ravenna is tan–1(7376).= 0.765 radians or 43.8◦north of east.Interactive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 0.5: Examples 1 through 7‡The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sectionsof the textbook for the