# Enrichment, Extension, and Application

1 point

## 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.
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 $$r \lt \sqrt{2}$$.
7. If $$\sin(\alpha) + \cos(\alpha) = 1.2$$, find the value of $$\sin^3(\alpha) + \cos^3(\alpha)$$.
8. 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 $$BC$$.