# Tag Archives: Trigonometric Properties 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} […] # 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 […] 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 […] # Integration using Trigonometric Properties

Trigonometric properties such as the sum of squares of sine and cosine with the same angle is one, $$\displaystyle \sin^2{\theta} + \cos^2{\theta} = 1 \\ \cos\Big(\frac{\pi}{2} – \theta \Big) = \sin{\theta}$$ can simplify harder integration. Worked Example of Integration using Trigonometric Properties (a)   Find $a$ and $b$ for \(\displaystyle \frac{1}{x(4-x)} = \frac{a}{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{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} […]