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Proof 1 \( \sin (\alpha-\beta) = \sin \alpha \cos \beta-\cos \alpha \sin \beta \) \( \require{AMSsymbols} \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} \\ &= \frac{QN}{OP}-\frac{PR}{OP} \\ &= \frac{QN}{\color{red}{ON}} \times \frac{\color{red}{ON}}{OP}-\frac{PR}{\color{red}{PN}} \times […]
Proof 1 \( \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \) \( \require{AMSsymbols} \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} \\ […]
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 \( z = \cos \theta + i \sin \theta \). \( […]
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 \\\cos^2 \theta &= […]
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 this 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 equation of the unit circle is $x^2+y^2=1$.\( […]
You should know these terms relating to the parts of a circle. Centre of a Circle The centre of a circle is the point that is equidistant from all points on the circle. The centre of a circle \( x^2 + y^2 = 9 \) is \( (0,0) \).The centre of a circle \( (x-1)^2 […]
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 […]
Trigonometric properties such as the sum of squares of sine and cosine with the same angle are one,$$ \displaystyle \sin^2{\theta} + \cos^2{\theta} = 1 \\\cos\Big(\frac{\pi}{2}-\theta \Big) = \sin{\theta} $$can simplify harder integration. Worked on an Example of Integration using Trigonometric Properties (a) Find \(a\) and \(b\) for \(\displaystyle \frac{1}{x(4-x)} = \frac{a}{x} + \frac{b}{4-x} \). […]
There are many areas to apply the compound angle formulas, and trigonometric proof using the compound angle formula is one of them. $$ \begin{aligned} \require{AMSsymbols} \require{color}\sin (x + y) &= \sin x \cos y + \sin y \cos x &\color{green} (1) \\\sin (x-y) &= \sin x \cos y-\sin y \cos x &\color{green} (2) \end{aligned} $$Using […]