Proof 1 \( \sin (\alpha – \beta) = \sin \alpha \cos \beta – \cos \alpha \sin \beta \) \( \begin{align} \angle RPN &= 90^{\circ} – \angle PNR \\ &= \angle RNB \\ &= \angle QON \\ &= \alpha \\ \sin(\alpha – \beta) &= \sin \angle MOP \\ &= \displaystyle \frac{MP}{OP} \\ &= \frac{MR-PR}{OP} \\ &= […]

# Tag Archives: Trigonometric Properties

# Trigonometric Ratios of Sums of Two Angles

Proof 1 \( \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \) \( \begin{align} \angle RPN &= 90^{\circ} – \angle PNR \\ &= \angle RNO \\ &= \angle RNO \\ &= \angle NOQ \\ &= \alpha \\ \sin (\alpha + \beta) &= \sin \angle AOC \\ &= \displaystyle \frac{MP}{OP} […]

# Powers of Cosine or Sine by Complex Number

Question 1 Show that \( \cos 4 \theta = 8 \cos^4 \theta – 8 \cos^2 \theta + 1 \). (a) By considering the real part of \( z^4 \), prove \( \cos 4\theta = \cos^4 \theta – 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta \) by letting \( z = \cos \theta + i […]

# 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-Radian Conversions

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 […]

# Integration using Trigonometric Properties

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

There are many areas to apply the compound angle formulas, and trigonometric proof using the 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 &\color{green} […]