RECIPROCAL TRIGONOMETRIC FUNCTIONS AND THEIR GRAPHS (Investigation) DATE: Part A: Using the table below, along with the graphs of the primary trig. functions: 1. Indicate where the function is increasing (T ) and decreasing (V) within the specified domain. 2. Determine the value of y at each of the quadrantal angles, - -, 0, 5 , n, ", 2n. 3. Starting with y = cscx = - sinx To find the behaviour of cscx, we reciprocate the results found in column 2. That is if sinx is increasing from 0 to , then csex is in that same interval. If sin0 = 0 then csco = which implies Fill in the 5th column with your findings then sketch the graph on the same grid as y = sinx within the domain, x 6 -7, 2x] 4. Repeat step 3 for y = secx and y = cotx. PRIMARY FUNCTIONS RECIPROCAL FUNCTIONS y = sinx y = COSX y = tanx y = CSCX y = secx y = cotx xE (-4,0) 0 x E (0, =)GRAPHS OF PRIMARY AND RECIPROCAL TRIGONOMETRIC FUNCTIONS Part B: Summarize the properties of the trigonometric functions below using the information you obtained from your graphs and the chart above. Domain: 2 . Range: Period VA: -2+ f(x) >0: (1 cycle) Intervals of (1 cycle) Domain: Range: Period VA: -2- f(x) >0: (1 cycle) Intervals of (1 cycle) Domain: Range: 2- Period VA: -2x f(x) >0: -2 (1 cycle) Intervals of (1 cycle) MHF4UP - Section 5.2 2PRACTICE: Graph the following functions with the domain, x 6 [-2x, 2x]. State any key characteristics for each trig function. y = sec x - 2 y = cot (x - =) y = csc (x +#)+3 y = sec3x -2-Application: 1. When the Sun is directly overhead, its rays pass through the atmosphere as shown. Call this 1 unit of atmosphere. When the Sun is not over head but is inclined at angle x to the surface of Earth, its rays pass thorough more air before they reach sea level. Call this y unit of atmosphere. The value of y affects the temperature at the surface of Earth. Atmosphere a) Use the diagram to determine an expression for y in term of angle x. y Sea Level b) Describe what happens to the value of y as x approaches 0. Why does this make sense? CONNECTIONS Many factors affect the temperature at the surface of Earth, such as particulate pollution, winds, and cloud cover Climate scientists take these into account and build mathematical models that are used to predict temperatures. Since there are so many Note: This model makes the assumption that the surface of the Earth is assumed to variables, many of which be flat. The atmosphere is assumed to be uniform and to end at a particular height are difficult to model, the above the surface of the earth. temperature predictions lose accuracy quickly with time. 2. A rotating light on top of a lighthouse sends out rays of light in opposite directions. As the beacon rotates, the rayat angle makes a spot of light light that moves along the shore. The lighthouse is located 500 m from the ray shoreline and makes one complete rotation every 2 min. lighthouse beacon a) Determine the equation that expresses the distance, d, in metres, as a funciton of time, t, in minutes. 500 m shore b) Graph the function in part a) and explain the significance of the asymptote in the graph at 0= =