Algebra Concepts and Expressions Review

10 A piece of wire of length $66 \mathrm{~cm}$ is bent to form the five sides of a pentagon. The pentagon consists of three sides of a rectangle and two sides of an equilateral triangle. The sides of the rectangle measure $x \mathrm{~cm}$ and $y \mathrm{~cm}$ and the sides of the triangle measure $x \mathrm{~cm}$, as shown in the diagram below. 10 (a) (i) You are given that $\sin 60^{\circ}=\frac{\sqrt{3}}{2}$ Explain why the area of the triangle is $\frac{\sqrt{3}}{4} x^{2}$ [1 mark] 10 (a) (ii) Show that the area enclosed by the wire, $A \mathrm{~cm}^{2}$, can be expressed by the formula \[ A=33 x-\frac{1}{4}(6-\sqrt{3}) x^{2} \] [3 marks]

Given the polynomial \( P(x) = 2x^3 - 3x^2 + 4x - 5 \) and \( Q(x) = x - 1 \), find the remainder when \( P(x) \) is divided by \( Q(x) \). Additionally, if \( R(x) \) is the remainder and \( \theta \) is the root of \( Q(x) \), find the value of \( \cos(\theta) \) if \( R(\theta) = \cos(\theta) \)

Solve for \( x \) in the equation \( \frac{\cos(x)}{\cos(x) - \sin(x)} = 2 \)

Solve the following trigonometric equation for \( x \): \[2\sin(x)\cos(x)+\sin(x)-1=0\]

If \(\sin^2{x} - \cos^2{x} = 1\), find the value of \(x\).

Find the value of \( x \) if \( \cos(2x) = \frac{1}{2} \) and \( x^3 + 1 = 0 \)

Given two algebraic expressions \(4x^2 \cos^2 y + 9x^2 \sin^2 y\) and \(5x^2 \cos^2 y + 2x^2 \sin^2 y\), find the greatest common factor (GCF) and simplify the expressions.

If \( \sin(x) = a \) and \( \cos(x) = b \), express the expression \( 2\sin(x)\cos(x) - 2a^2 + 2b^2 \) in its simplest form.

Solve the equation \(2\sin^2x - 3\sin x - 2 = 0\) in the interval \([0, 2\pi)\).

Find the domain of the function \( y = \sqrt{\sin(x)} + \frac{1}{x-3} \).

If \(x = \frac{\pi}{6}\), find the value of \(\frac{2cosx + sinx}{sinx}\)

Evaluate the expression \(2 cos(x) + 3 sin(y)\) given that \(cos(x) = \frac{1}{2}\) and \(sin(y) = \frac{3}{5}\).

Given that \( x = \sqrt[3]{27} \) and \( y = \cos(\frac{\pi}{3}) \), solve for \( z = 3x^2 - 2y \).

Find the holes in the graph of the function \( f(x) = \frac{\sin{x} - 1}{x^2 - 1} \).