In the diagram, the graphs of $f(x) = 2 ext{sin}2x$ and $g(x) = - ext{cos}(x + 45^ ext{o})$ are drawn for the interval $x ext{ in } [0^ ext{o} ; 180^ ext{o}]$ - NSC Mathematics - Question 6 - 2023 - Paper 2

The function $g(x) = - ext{cos}(x + 45^ ext{o})$ achieves its maximum when $ext{cos}(x + 45^ ext{o})$ is minimized, and its minimum when $ext{cos}(x + 45^ ext{o})$ is maximized. The range of $ext{cos}$ is $[-1, 1]$, thus,

• The maximum value of $g$ is 0.
• The minimum value of $g$ is -1.

Therefore, the range of $g$ is:

$g(x) ext{ in } [-1, 0]$

To solve $f(x) imes g(x) > 0$, we need to analyze the signs of both functions in the specified interval. Since both functions will be positive or both negative for the product to be positive:

1. For $f(x) > 0$, we find:

   • From $f(x) = 2 ext{sin}2x > 0$, we have:
   • $2x$ must be in the first or second quadrant. Thus:
   • This occurs when $0 < 2x < 180^ ext{o}$ or $360^ ext{o} < 2x < 540^ ext{o}$.
   • Therefore, this simplifies to:
   • $0 < x < 90^ ext{o}$.

2. For $g(x) > 0$, we analyze:

   • $g(x) = - ext{cos}(x + 45^ ext{o}) > 0$ or $ext{cos}(x + 45^ ext{o}) < 0$. This implies that:
   • $x + 45^ ext{o}$ is in the second quadrant. Thus, $90^ ext{o} < x + 45^ ext{o} < 180^ ext{o}$, yielding:
   • $45^ ext{o} < x < 135^ ext{o}$.

Combining intervals, the solution for $f(x) imes g(x) > 0$ is:

$x ext{ in } (45^ ext{o}, 90^ ext{o}) ext{ or } (105^ ext{o}, 135^ ext{o})$

To solve $f(x) + 1 ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } ext{ } = 0$, we rearrange:

$f(x) ext{ } ext{ } + 1 ext{ } ext{ } ext{ } ext{ } = 0 ext{ } ext{ } ext{ } ext{ } = 2 ext{sin}2x + 1 ext{ } = 0$

Thus, we have:

$2 ext{sin}2x = -1$

This simplifies to:

ext{sin}2x = -rac{1}{2}

The general solution is:

$2x = 210^ ext{o} + 360^ ext{o}k ext{ and } 2x = 330^ ext{o} + 360^ ext{o}k ext{ for integers } k$

From this we get:

$x = 105^ ext{o} + 180^ ext{o}k ext{ and } x = 165^ ext{o} + 180^ ext{o}k$

Within the interval $[0^ ext{o}, 180^ ext{o}]$:

• For $k=0, x=105^ ext{o}$ and $x=165^ ext{o}$.

Therefore:

$x ext{ in } [105^ ext{o}, 165^ ext{o}]$

Since $p(x) = -f(x)$ and $D(k, -1)$ lies on $p$, we set $-f(k) = -1$. Thus we have:

$-2 ext{sin}2k = -1$

This simplifies to:

ext{sin}2k = rac{1}{2}

The general solutions for this equation are:

$2k = 30^ ext{o} + 360^ ext{o}k ext{ or } 2k = 150^ ext{o} + 360^ ext{o}k$

This gives us:

• $k = 15^ ext{o} + 180^ ext{o}n$
• $k = 75^ ext{o} + 180^ ext{o}n$

For integer values of $n$ and considering the interval $[0^ ext{o}, 180^ ext{o}]$, we find:

• For $n=0$, $k = 15^ ext{o}$ and $k = 75^ ext{o}$.

Hence the required value(s) are:

$k ext{ in } [15^ ext{o}, 75^ ext{o}]$

To determine the equation of $h$, we start with the original function for $g(x)$:

$g(x) = - ext{cos}(x + 45^ ext{o})$

When $g$ is translated $45^ ext{o}$ to the left, the transformation can be expressed as:

$h(x) = g(x + 45^ ext{o})$

Thus,

$h(x) = - ext{cos}((x + 45^ ext{o}) + 45^ ext{o})$

Simplifying this gives:

$h(x) = - ext{cos}(x + 90^ ext{o})$

Using the trigonometric identity, we find:

$h(x) = ext{sin}x$

Therefore, the equation of $h$ in its simplest form is:

$h(x) = ext{sin} x$