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Exam-Style Questions.Problems adapted from questions set for previous Mathematics exams. |
1. | IB Analysis and Approaches |
(a) Show that:
$$ \cos 2x - \sin 2x + 1 = 2 \cos x ( \cos x - \sin x) $$(b) Hence or otherwise, solve the following equation for \( \pi \lt x \lt 3\pi \).
$$ \cos 2x - \sin 2x + 1 = \sin x - \cos x $$2. | IB Analysis and Approaches |
(a) Show that the equation \( 2 \sin^2 x - 5 \cos x = -1\) may be written in the form \( 2 \cos^2 x + 5 \cos x = 3\)
(b) Hence, solve the equation \( 2 \sin^2 x - 5 \cos x = -1 \), \( 2\pi \lt x \lt 4\pi \).
3. | A-Level |
The cosine of acute angle \( \alpha \) is \( \frac{1}{ \sqrt 5} \)
The angle \( \beta \) is obtuse and \( \sin \beta = \sqrt \frac{2}{3} \).
(a) Find exact values of \( \tan \alpha \) and \( \tan \beta \).
(b) Hence show that \( \tan( \alpha - \beta ) \) can be written as \(a+b \sqrt 2 \) where \(a\) and \(b\) are rational numbers4. | IB Analysis and Approaches |
Consider the functions \(f(x)=\sin(x) \text{ and } g(x) = \tan(x+\frac{\pi}{2})\) in the region where \(\frac{\pi}{2}\le x \le \pi\)
The graphs \(y=f (x)\) and \(y = g(x)\) intersect at a point A whose x-coordinate is \(a\).
(a) Show that \(\sin^2{a}=-\cos{a}\).
(b) Hence, show that the tangents to the curves at A intersect at right angles.
(c) Find the value of \(\cos(a)\). Give your answer in the form \( \dfrac{a + \sqrt{b}}{c} \).
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