Module 5: Lesson 2

 

Self-Check 2

3. Questions 5, 16, and 18 on pages 275 to 279 of the textbook
   5.  
      1. A graphical solution is shown.
         
         
          
         
         
         x ≈ 1.33
      2. An algebraic solution is shown.
         
         
          
         
         
         
          
         
         
         363.590…° is not in the domain 0° < x ≤ 360°. However, the coterminal angle 363.590…° − 360° = 3.590…°.
         
         So, x ≈ 3.59° and 86.41°.
      3. An algebraic solution is shown.
         
         
          
         
         
         On the domain 0 < 2x − 5 ≤ 2 π,
         
         
          
         
         
         x ≈ 3.093 + nπ, n ∈ I, x ∈ R and x ≈ 5.048 + nπ, n ∈ I, x ∈ R.
      4. A graphical solution is shown.
         
         
          
         
         
         The first two solutions occur at x ≈ 4.5° and 7.5°. Because the period is 8°, the general equation is x ≈ 4.5°, 7.5° ± (8°) n, n ∈ I.
   16.  
       1. 
          Adapted from:  Pre-Calculus 12. Whitby, ON:
          McGraw-Hill Ryerson, 2011. Reproduced with permission. 
          
       2. The temperature that is halfway between the maximum average monthly temperature and the minimum average monthly temperature in Winnipeg can be found by performing the following operation:
          
          
           
          
       3. The a value can be determined by finding the amplitude:
          
          This means that the amplitude is
          
          The graph appears to be a sinusoidal graph (of cosine) that has been reflected about the x-axis; this means that you can state that the a value is −18.1. If this is done, there will be a phase shift of 1 unit to the right (because the lowest temperature occurs in January, or the first month). The period is about 12 months. This means that the b value is The vertical displacement is
          
          Therefore, an approximate equation for the sinusoidal function is In this formula, x represents the time, in months, and y represents the average monthly temperature, in degrees Celsius, for Winnipeg.
       4.  
          Adapted from: Pre-Calculus 12. Whitby, ON:
          McGraw-Hill Ryerson, 2011. Reproduced with permission. 
          
       5.  
          
          The two intersection points occur at (5.76, 16) and (8.24, 16). This means that the average temperature will be above 16°C for 8.24 − 5.76, or for approximately 2.5 months.
   18.  
       1. 
          Adapted from Pre-Calculus 12. Whitby, ON: McGraw-Hill Ryerson, 2011.
          Reproduced with permission.
          
          
          
       2. The a value can be determined by finding the amplitude:
          
          This means that the amplitude is you can state that the a value is 10. If this is done, there will be a phase shift of 0.45 units to the left. The period is 3.3 s − 0.3 s or 3 s. Thus, the b value is The vertical displacement is
          
          Thus, an approximate equation for the sinusoidal function is In this formula, t represents the time, in seconds, and y represents the height of the mass, in centimetres, above the floor.
       3. To do this, you can substitute 17.2 in place of t.
          
          
           
          
       4. 

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