Trigonometry and Basic Mathematics for Calculus

These flashcards cover fundamental trigonometry, coordinate geometry, and numerical sequences (AP/GP) as presented in the lecture notes.

Sine ($\sin(\theta)$) Ratio

The ratio of the Perpendicular ($P$) to the Hypotenuse ($H$) in a right-angled triangle.

Cosine ($\cos(\theta)$) Ratio

The ratio of the Base ($B$) to the Hypotenuse ($H$) in a right-angled triangle.

Tangent ($\tan(\theta)$) Ratio

The ratio of the Perpendicular ($P$) to the Base ($B$) in a right-angled triangle.

Inverse Trigonometric Relations

Pairs of ratios where the product equals $1$, specifically: $\sec(\theta) \times \cos(\theta) = 1$, $\sin(\theta) \times \csc(\theta) = 1$, and $\cot(\\theta) \times \tan(\theta) = 1$.

Pythagorean Identity (Sine and Cosine)

Fundamental trigonometric identity derived from Pythagoras Theorem: $\sin^2(\theta) + \cos^2(\theta) = 1$.

Trigonometric Range

The output values for sine and cosine range from $-1$ to $+1$, while tangent ranges from $-\infty$ to $+\infty$.

Special Angle Values ($37^\circ$)

$\sin(37^\circ) = \frac{3}{5}$ and $\cos(37^\circ) = \frac{4}{5}$.

Special Angle Values ($53^\circ$)

$\sin(53^\circ) = \frac{4}{5}$ and $\cos(53^\circ) = \frac{3}{5}$.

Arc Length Formula

The length of an arc calculated as $\text{Radius} \times \theta$, where $\theta$ must be in radians.

Radian

The SI unit of an angle, where $180^\circ = \pi$ radian.

Orthogonal Lines

Lines that meet at an angle of $\theta = 90^\circ$.

Complementary Angles

Two angles whose sum is $\alpha + \beta = 90^\circ$.

Supplementary Angles

Two angles whose sum is $\alpha + \beta = 180^\circ$.

Half Angle Formula (Sine)

$\sin(\alpha) = 2\sin\left(\frac{\alpha}{2}\right)\cos\left(\frac{\alpha}{2}\right)$.

ASTC Rule

A rule determining the sign of trigonometric functions in four quadrants: All positive in 1st, Sine/Cosec in 2nd, Tan/Cot in 3rd, and Cos/Sec in 4th.

Small Angle Approximation

When $\theta$ is less than $5^\circ$, $\sin(\theta) \approx \theta$, $\tan(\theta) \approx \theta$ (in radians), and $\cos(\theta) \approx 1$.

Phase Difference ($\Delta\phi$)

The angle difference between two trigonometric functions of the same type, calculated as $\Delta\phi = \phi_2 - \phi_1$.

Phasor Diagram

A representation of trigonometric functions using phases (angles).

Arithmetic Progression (AP)

A sequence of numbers where the difference between two consecutive numbers is constant.

Common Difference ($d$)

In an Arithmetic Progression, the value calculated as $\text{nth term} - (n-1)\text{th term}$.

Geometric Progression (GP)

A sequence where the ratio of any two consecutive numbers is constant.

Common Ratio ($r$)

In a Geometric Progression, the value calculated as $\frac{\text{nth term}}{(n-1)\text{th term}}$, used to find terms using $a \times r^{(n-1)}$.

Distance Formula (2D)

The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ calculated as $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.

Slope ($m$)

The tangent of the angle $\theta$ ($m = \tan(\theta)$) representing the ratio of the change in the $y$-axis to the change in the $x$-axis ($\frac{\Delta y}{\Delta x}$).

Equation of a Straight Line

$y = mx + c$, where $m$ is the slope and $c$ is the intercept on the $y$-axis.

Rectangular Hyperbola

The graph formed by an inversely proportional relation $y \propto \frac{1}{x}$ or $xy = \text{constant}$, where the graph never cuts the axes.

Parabolic Graph

The graph formed by the relation $y = x^2$, typically showing an upward opening curve.