Tag Archives: Trigonometric Properties

Trigonometric Ratios of Differences of Two Angles

Trigonometric Ratios of Differences of Two Angles

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 […]

Trigonometric Ratios

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 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 $x^2+y^2=1$.\( […]

Degree-Radian Conversions

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 one […]

Integration using Trigonometric Properties

Integration using Trigonometric Properties

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 \(\displaystyle \frac{1}{x(4-x)} = \frac{a}{x} + \frac{b}{4-x} \). […]