Enrichment, Extension, and Application

Question 1

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Exercises

  1. For what values of \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \) does the equation \( x^2 + \left( 2\sin(\theta) \right)x + \cos(2\theta) = 0 \) have real roots?
    1. Prove that \[ \tan(\theta) + \cot(\theta) = \dfrac{2}{\sin(2\theta)} \]
    2. In the right-angled triangle \( ABC \), \( AC = 4 \), \( BP = 1 \), and \( BP \perp AC \). Find a value for angle \( \theta = \angle{ACB} \). Triangle ABC with right angle at B, point P on AC connected to B, and angle ACB equal to theta
  2. Find all values of \( x \) with \( 0 \leq x \leq 2\pi \) such that \[ \sin(2x) + \cos(2x) + \sin(x) + \cos(x) + 1 = 0 \]
    1. In the diagram, the line segment \( OP \) makes an angle of \( \alpha \) radians with the \( x \)-axis. The line segment \( OP \) is then rotated by an angle of \( \theta \) radians counterclockwise about the origin such that the point \( P \) is mapped onto the point \( P' \), with \( \left\lvert OP \right\rvert=\left\lvert OP' \right\rvert \). If the coordinates of \( P \) are \( (x, y) \) and the coordinates of \( P' \) are \( (u, v) \), show that \[ \Big(u, v \Big) = \Big( x \cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta) \Big) \] Diagram of situation described in question
    2. Let two points, \( P(a, b) \) and \( Q(c, d) \), both be rotated counterclockwise about the origin through the same angle \( \theta \) (in radians). If \( P'(a', b') \) and \( Q'(c', d') \) are the images of the points \( P \) and \( Q \) after the rotation, show that the lengths \( \left\lvert PQ \right\rvert \) and \( \left\lvert P'Q' \right\rvert \) are the same, and hence show that lengths of line segments are invariant under rotation about a fixed point.
    3. In the diagram, the points \( P \), \( Q \), and \( R \) are rotated counterclockwise about the origin through the same angle \( \theta \) radians. If \( P' \), \( Q' \), and \( R' \) are the images of \( P \), \( Q \), and \( R \) respectively after the rotation, show that \( \angle{PQR} = \angle{P'Q'R'} \), and hence show that angles between line segments are invariant under rotation about a fixed point. Diagram of situation described in question
  3. If \( \sqrt{6}\sin(\theta) + \sqrt{2}\cos(\theta) = 2 \) for \( 0 \leq \theta \leq 2\pi \), find \( \cos(2\theta) \).
  4. Find all values of \( \theta \) with \( 0 \leq \theta \leq 2\pi \) such that \( \cos(2\theta) = \cos(\theta) + \sin(\theta) \).
  5. In a certain triangle \( ABC \), \( \cos(A)\cos(B) + \sin(A)\sin(B)\sin(C) = 1 \). Determine all possible values for \( \angle{C} \).
    1. If \( \theta \neq \dfrac{k\pi}{2} \), where \( k \) is an integer, prove that \[ \cot(\theta) - \cot(2\theta) = \dfrac{1}{\sin(2\theta)} \]
    2. Without a calculator, find the angle \( \theta \) such that \[ \dfrac{1}{\sin(8^\circ)} + \dfrac{1}{\sin(16^\circ)} + \dfrac{1}{\sin(32^\circ)} + \ldots + \dfrac{1}{\sin(4096^\circ)} + \dfrac{1}{\sin(8192^\circ)} = \dfrac{1}{\sin(\alpha)} \] Note: The sum involves terms of the form \( \dfrac{1}{\sin\left( \left( 2^k \right)\left( 8^\circ \right)\right)} \) for \( k = 0, 1, \ldots, 10 \).
  6. Without using a calculator, determine positive integers \( m \) and \( n \) for which \[ \sin^6(1^\circ) + \sin^6(2^\circ) + \sin^6(3^\circ) + \ldots + \sin^6(87^\circ) + \sin^6(88^\circ) + \sin^6(89^\circ) = \frac{m}{n} \] Note: The sum involves terms of the form \( \sin^6{\left( k^\circ \right)} \) for \( k = 1, 2, \ldots, 10 \).
  7. As in the diagram, point \( B \) lies on side \( AC \) of \( \triangle{ACD} \) and point \( E \) lies on \( CD \) extended. Suppose that \( \angle{ACD} = \alpha \), \( \angle{BDA} = 2\theta \), and \( \angle{BDC} = \angle{ADE} = \beta \). Prove that \( AB = BC \) if and only if \( \dfrac{\cos(\alpha - \theta)}{\cos(\alpha + \theta)} = 2 \). AC meets CE at C, DB meets AC at B, AD meets CE at D, angles labelled as described in question
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