In the diagram, the graphs of $f(x) = -3 ext{sin} rac{x}{2}$ and $g(x) = 2 ext{cos}(x - 60^ op)$ are drawn in the interval $x ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ }$ $[-180^ op; 180^ op]$ - NSC Mathematics - Question 6 - 2018 - Paper 2
Question 6
In the diagram, the graphs of $f(x) = -3 ext{sin} rac{x}{2}$ and $g(x) = 2 ext{cos}(x - 60^ op)$ are drawn in the interval $x ext{ } ext{ } ext{ } ext{ } ext{... show full transcript
Worked Solution & Example Answer:In the diagram, the graphs of $f(x) = -3 ext{sin} rac{x}{2}$ and $g(x) = 2 ext{cos}(x - 60^ op)$ are drawn in the interval $x ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ }$ $[-180^ op; 180^ op]$ - NSC Mathematics - Question 6 - 2018 - Paper 2
Step 1
Write down the period of $f$
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Answer
The period of a sine function is given by the formula:
ext{Period} = rac{2 ext{π}}{k}
where k is the coefficient of x. In this case, the function f(x) = -3 ext{sin} rac{x}{2} has k = rac{1}{2}, hence,
ext{Period} = rac{2 ext{π}}{ rac{1}{2}} = 4 ext{π}.
Step 2
Write down the range of $g$
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Answer
The function g(x)=2extcos(x−60op) has a range that can be derived from its amplitude. The amplitude is 2, so the range of g is:
[−2,2].
Step 3
Calculate $f(p) - g(p)$
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Answer
To calculate f(p)−g(p), substitute the value of p into both functions:
Calculate f(p) = -3 ext{sin} rac{p}{2}.
Calculate g(p)=2extcos(p−60op).
Thus, the difference is:
f(p) - g(p) = -3 ext{sin} rac{p}{2} - 2 ext{cos}(p - 60^ op).
Step 4
Use the graphs to determine the value(s) of $x$ in the interval $[-180^ op; 180^ op]$ for which: g(x) > 0
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From the graph of g(x), identify intervals where the curve is above the x-axis. Based on the graph, the values of x for which g(x)>0 are found to be:
Between −60op and 60op.
Step 5
Use the graphs to determine the value(s) of $x$ in the interval $[-180^ op; 180^ op]$ for which: f(x) > 0
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Answer
From the graph of f(x), determine the intervals where the curve is above the x-axis. The values of x satisfying f(x)>0 include:
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