Trigonometry
Mathematical Induction involving Compound Angle Formula of Tangent
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 […]
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 […]
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