# Tag Archives: Trigonometric Properties 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 of Sums 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 RNO \\ &= \angle RNO \\ &= \angle NOQ \\ &= \alpha \\ \sin (\alpha + \beta) &= \sin \angle AOC \\ &= \displaystyle \frac{MP}{OP} \\ […] # 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 \\\cos^2 \theta &= […] # 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 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 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 $\displaystyle \frac{1}{x(4-x)} = \frac{a}{x} + \frac{b}{4-x}$. […] # 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{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 […]