Let $z = \cos \theta + i \sin \theta$. (a)   By considering the real part of $z^4$, prove $\cos 4\theta = \cos^4 \theta - 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta$. \begin{align} \displaystyle \require{color} z &= \cos \theta + i \sin \theta \\ z^4 &= (\cos [...] # Applications of the Unit Circle The identify \cos^2 \theta + \sin^2 \theta = 1 is required for finding trigonometric ratios. Example 1 Find exactly the possible values of \cos \theta for \sin \theta = \dfrac{5}{8}. \( \begin{align} \displaystyle \cos^2 \theta + \sin^2 \theta &= 1 \\ \cos^2 \theta + \sin^2 \dfrac{5}{8} &= 1 \\ \cos^2 \theta + \dfrac{25}{64} &= 1 [...] # Trigonometric Ratios Circles with Cnetre (0,0) Consider a circle with centre (0,0) and radius r units. Suppose (x,y) is any point on this circle. Using ths distance formula; \( \begin{align} \displaystyle \sqrt{(x-0)^2+(y-0)^2} &= r \\ \therefore x^2+y^2 &= r^2 \end{align} $x^2+y^2 = r^2$ is the equation of a circle with centre $(0,0)$ and radius $r$. The [...] # Arc Length and Sector Area

You should know with these terms relating to the parts of a circle. The centre of a circle is the point which is equidistant from all points on the circle. A radius of a circle a straight line joining the centre of a circle to any point on the circumference. A minor arc is an [...] Degree Measurement of Angles One full revolution makes an angle of $360^{\circ}$, and the angle on a straight line is $180^{\circ}$. Therefore, one degree, $1^{\circ}$, can be defined as $\dfrac{1}{360}$ of one full revolution. For greater accuracy we define one minute, $1'$, as $\dfrac{1}{60}$ of one degree and one second, $1^{\prime \prime}$, as $\dfrac{1}{60}$ of [...] One of the most effective ways to solve Three Dimensional Trigonometry questions is to list all of the trigonometric ratios appeared in the questions. Then connect each relevant ratios to the question. Worked Examples of Three Dimensional Trigonometry Question 1 $X$ and $Y$ are two points on the same bank of a straight river and [...] 