Trigonometry

Rewrite Expressions in Sine and Cosine involving Powers of Cosine or Sine by Complex Number

October 17, 2019 0 comments

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

February 14, 2019 2 comments

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

February 11, 2019 0 comments

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

February 9, 2019 0 comments

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-Radian Conversions

February 8, 2019 0 comments

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 [...]

Three Dimensional Trigonometry

September 4, 2016 0 comments

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 [...]

Integration using Trigonometric Properties

August 24, 2016 0 comments

Trigonometric properties such as the sum of squares of sine and cosine with the same angle is one, $$ \displaystyle \sin^2{\theta} + \cos^2{\theta} = 1 \\ \cos\Big(\frac{\pi}{2} - \theta \Big) = \sin{\theta} $$ can simplify harder integration. Worked Example of Integration using Trigonometric Properties (a) Find $a$ and $b$ for \(\displaystyle \frac{1}{x(4-x)} = \frac{a}{x} [...]

Trigonometric Proof using Compound Angle Formula

August 19, 2016 0 comments

There are many areas to apply the compound angle formulas, and trigonometric proof using compound angle formula is one of them. $$ \begin{aligned} \require{color} \sin (x + y) &= \sin x \cos y + \sin y \cos x &\color{green} (1) \\ \sin (x - y) &= \sin x \cos y - \sin y \cos x [...]

3 Ways of Evaluating Nested Square Roots

August 18, 2016 2 comments

Nested square roots or nested radical problems are quite interesting to solve. The key skill for this question is to understand how the students can handle "...". This enables us to setup a quadratic equation to evaluate its exact value using the quadratic formula, $\require{color}$ $$x= \frac{-b \ \pm \sqrt{b^2 - 4ac}}{2a}$$. Let's take a [...]

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