Write down the range of \(h\).
About this question
Question 7.4 of Mathematics Grade 12 Paper 2, NSC Examination (DBE, November 2025). It tests the Term 1 ATP topic Trigonometry (Compound Angles, Identities, Equations).
- TermTerm 1
- ATP topicTrigonometry (Compound Angles, Identities, Equations)
- PaperPaper 2
- Marks1 mark
The graph of \(f(x) = \cos 2x\) is drawn for \(x \in [-135^\circ\,;\,135^\circ]\).
Diagram: \(f\) starts at 0 at \(x = -135^\circ\), has minimum \(-1\) at \(x = -90^\circ\), cuts the \(x\)-axis at \(-45^\circ\), reaches its maximum 1 at \(x = 0^\circ\), cuts the \(x\)-axis at \(45^\circ\), has minimum \(-1\) at \(90^\circ\) and returns to 0 at \(135^\circ\). Gridlines are at every \(45^\circ\).
\(y \in [-1\,;\,1]\), that is \(-1 \leq y \leq 1\)
More Questions On Trigonometry (Compound Angles, Identities, Equations)
The identity is undefined where \(\cos x = 0\) (\(\tan x\) is undefined and the RHS has \(\cos x\) in the denominator).
\[x = 90^\circ + k \cdot 360^\circ \text{ or } x = 270^\circ + k \cdot 360^\circ,\ k \in \mathbb{Z}\]\(x = -90^\circ\) or \(x = 90^\circ\)
Divide by \(\cos x\) (\(\cos x \neq 0\), since \(\cos x = 0\) gives \(\sin x = \pm 1\), which is not a solution):
\[2\cos^2 x = \tan x\]\[2\cos^2 x - 1 + 1 = \tan x\]\[\cos 2x + 1 = \tan x, \text{ so } g(x) = f(x)\]The graphs intersect at \(x = 45^\circ\) and \(x = -135^\circ\) (\(180^\circ\) apart).
\(x = 45^\circ + k \cdot 180^\circ,\ k \in \mathbb{Z}\)
Write \(4x = 7x - 3x\) and \(10x = 7x + 3x\):
\[\sin 4x - \sin 10x = \sin(7x - 3x) - \sin(7x + 3x)\]\(= -2\cos 7x\sin 3x\)
\(\tan x \leq 0\) where the graph is on or below the \(x\)-axis.
\(x = -180^\circ\) or \(-90^\circ \lt x \leq 0^\circ\)
\(f\) is on or below the \(x\)-axis between its \(x\)-intercepts \(\pm 120^\circ\) and the ends of the interval.
\(-180^\circ \leq x \leq -120^\circ\) or \(120^\circ \leq x \leq 180^\circ\)
So \(g\) and \(f\) must have opposite signs (or \(g = 0\)).
From the graphs, this happens for \(0^\circ \leq x \leq 22{,}5^\circ\) and for \(112{,}5^\circ \leq x \lt 135^\circ\) (at \(45^\circ\) and \(135^\circ\), \(\tan 2x\) is undefined).
\(x \in [0^\circ\,;\,22{,}5^\circ] \cup [112{,}5^\circ\,;\,135^\circ)\)