Enrichment, Extension, and Application

1 point

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 $$\theta = \angle{ACB}$$.
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 $\Big(u, v \Big) = \Big( x \cos(\theta) - y \sin(\theta), x \sin(\theta) + y \cos(\theta) \Big)$
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.
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 $$\dfrac{\cos(\alpha - \theta)}{\cos(\alpha + \theta)} = 2$$.