Trigonometric properties and formulae can be used to perform proofs using mathematical induction. In this example, we use the following compound angle formulae for mathematical induction. $$ \displaystyle \begin{align} \tan (\alpha + \beta) &= \frac{\tan \alpha + \tan \beta}{1 – \tan \alpha \tan \beta} \\ \tan (\alpha – \beta) &= \frac{\tan \alpha – \tan \beta}{1 […]

# Trigonometry

# Integrating Trigonometric Functions by Recognition

Example 1 Find the derivative of \( \sin (2x-5) \) and use this result to deduce \( \displaystyle \int 10 \cos (2x-5) dx \). \( \begin{align} \displaystyle \frac{d}{dx} \sin (2x-5) &= \cos (2x-5) \times \frac{d}{dx} (2x-5) &\color{green}{\text{Chain Rule}} \\ &= \cos (2x-5) \times 2 \\ &= 2 \cos (2x-5) \\ \sin (2x-5) &= \int 2 […]

# Trigonometric Equation involving Double Angle and Compound Angle Formula

Trigonometric Equation involving Double Angle and Compound Angle Formula

# Definite Integration of Trigonometric Functions using Substitution

Find exact value of definite integration of trigonometric using substitution. It is useful to utilise the property of integral of even function, as well as reverse the of integral limits by chaning its sign.

# Trigonometric Ratios of Differences 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 RNB \\ &= \angle QON \\ &= \alpha \\ \sin(\alpha – \beta) &= \sin \angle MOP \\ &= \displaystyle \frac{MP}{OP} \\ &= \frac{MR-PR}{OP} \\ &= […]

# 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 […]