Enrichment, Extension, and Application

Question 1

1 point

Print

Exercises

  1. Determine the constants \( a \) and \( b \) so that \[ \frac{-3 + 4\cos^2{(\theta)}}{1 - 2\sin{(\theta)}} = a+b \sin{(\theta)} \] for all values of \( \theta \).
  2. Prove there are no real values of \( x \) such that \( 2 \sin(x) = x^2 - 4x + 6 \).
  3. If \( f(x) = \sin^2(x) - 2\sin(x) + 2 \), find the minimum and maximum values of \( f(x) \).
  4. Find the number of solutions to each of the following equations.
    1. \( x = 10 \cos(x) \)
    2. \( 3 \pi \big(1 - \cos(x)\big) = 2x \)
    3. \( \cos(x) = \log_{3\pi} (x) \)
  5. A rectangle is inscribed between the graphs of \( y = 3 \sin{\left( \frac{x}{2} \right)} \) and \( y= 3 \cos{\left( \frac{x}{2} \right)} \) and below the \(x\)-axis, as shown. If the length of \( AB \) is \( \frac{\pi}{3} \), find the area of the rectangle.
    AB is the width of the rectangle drawn between 3*sin(x/2) and 3*cos(x/2)
  6. For the given triangle \( ABC \), \( \angle C = \angle A + \frac{\pi}{3} \). If \( BC = 1 \), \(AC = r \), and \( AB=r^2 \), where \( r \gt 1 \), prove that \( r \lt \sqrt{2} \). Diagram as described in question with angle A labelled alpha an angle C labelled alpha plus pi/3
  7. If \( \sin(\alpha) + \cos(\alpha) = 1.2 \), find the value of \( \sin^3(\alpha) + \cos^3(\alpha) \).
  8. Triangle ABC with angle A=pi/7, angle C=3*pi/7, and angle CBD=2*pi/7 In triangle \( ABC \), point \( D \) is chosen on side \( AC \) such that \( CD=1 \) and \( DA=x \), with angles in radians as shown.
    1. Prove that \( x \) is a solution to the equation \( x^3+x^2-2x-1 = 0 \).
    2. Prove that \[ \cos\left( \frac{\pi}{7} \right) \cos\left( \frac{2\pi}{7} \right) \cos\left( \frac{3\pi}{7} \right) = \frac{1}{8} \] (Note: Evaluating on a calculator does not constitute a “proof”)
  9. Define \( f(x) = \sin^6(x) + \cos^6(x)+k \left(\sin^4(x)+\cos^4(x) \right) \) for some real number \( k \).
    1. Determine all real numbers \( k \) for which \( f(x) \) is constant for all values of \( x \).
    2. If \( k = -0.7 \), determine all solutions to the equation \( f(x) = 0 \).
    3. Determine all real numbers \( k \) for which there exists a real number \( c \) such that \( f(c) = 0 \).
  10. An equilateral triangle \( ABC \) has side length \( 2 \). A square, \( PQRS \), is such that \( P \) lies on \( AB \), \( Q \) lies on \( BC \), and \( R \) and \( S \) lie on \( AC \), as shown. The points \( P \), \( Q \), \( R \), and \( S \) move so that \( P\), \(Q \), and \( R \) always remain on the sides of the triangle and \( S \) moves from \( AC \) to \( AB \) through the interior of the triangle. If the points \( P\), \(Q\), \(R \), and \( S \) always form the vertices of a square, show that the path traced out by \( S \) is a straight line parallel to \( BC \). Diagram as described in question
© CEMC and University of Waterloo, Powered by Maplesoft