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 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-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 [...]
Volumes for Two Functions

Volumes for Two Functions

If the region bounded by the upper function $y_{upper}=f(x)$ and the lower funciton $y_{lower}=g(x)$, and the lines $x=a$ and $x=b$ is revolved about the $x$-axis, then its volume of revolution is given by: $$ \begin{align} \displaystyle V &= \int_{a}^{b}{\Big([f(x)]^2 - [g(x)]^2\Big)}dx \\ &= \int_{a}^{b}{\Big(y_{upper}^2 - y_{lower}^2\Big)}dx \end{align} $$ Example 1 Find the volume of revolution [...]
Area Between Two Functions

Area Between Two Functions

Home > iitutor If two functions $f(x)$ and $g(x)$ intersect at $x=1$ and $x=3$, and $f(x) \ge g(x)$ for all $1 \le x \le 3$, then the area of the shaded region between their points of intersection is given by: $$ \begin{align} \displaystyle A &= \int_{1}^{3}{f(x)}dx - \int_{1}^{3}{g(x)}dx \\ &= \int_{1}^{3}{\Big[f(x)-g(x)\Big]}dx \end{align} $$ Example 1 [...]