Higher Math 1st Paper - Chapter 06: Trigonometry (Notes by Md. Sabbir Hasan)

This set of flashcards covers vocabulary and key concepts from Chapter 06 of Higher Math 1st Paper, focusing on trigonometric angles, measurement systems, formulas for arc length and sector area, and the properties of trigonometric functions.

Geometric Angle

An angle formed when two rays intersect at a point, limited to a range between $0^\circ$ and $360^\circ$.

Trigonometric Angle

An angle generated by a ray rotating around its endpoint from its initial position, which can take any positive or negative value without limitation.

Positive Angle

An angle formed when a rotating ray moves in a counter-clockwise direction.

Negative Angle

An angle formed when a rotating ray moves in a clockwise direction.

Sexagesimal System

A system of angle measurement where one right angle equals $90^\circ$, $1^\circ$ equals $60'$ (minutes), and $1'$ equals $60''$ (seconds).

Radian

A constant angle subtended at the center of a circle by an arc whose length is equal to the radius of that circle.

Circular System

A system where angles are measured in radians ($^c$).

Relation between Degree and Radian

The conversion formula expressed as $\frac{R}{\pi} = \frac{D}{180}$, where $1^\circ = \frac{\pi}{180}$ radians and $1^c = \frac{180}{\pi} \approx 57.3^\circ$.

Arc Length ($s$)

The length of a circular arc calculated as $s = r\theta$, where $r$ is the radius and $\theta$ is the central angle in radians.

Area of a Sector

The area of a portion of a circle calculated as $\frac{1}{2}r^2\theta$ when $\theta$ is in radians, or $\frac{\theta}{360} \times \pi r^2$ when $\theta$ is in degrees.

Trigonometric Ratio: Sine

The ratio defined as $\frac{\text{Perpendicular}}{\text{Hypotenuse}}$.

Trigonometric Ratio: Cosine

The ratio defined as $\frac{\text{Base}}{\text{Hypotenuse}}$.

Quadrant Rule (2nd Quadrant)

The rule stating that between $90^\circ$ and $180^\circ$, only $\sin$ and $\csc$ are positive ($+$), while all other ratios are negative ($-$).

Co-functions

Pairs such as $\sin$ and $\cos$, $\tan$ and $\cot$, and $\sec$ and $\csc$ that are related by the identity $f(\theta) = g(90^\circ - \theta)$.

Fundamental Period

The smallest positive value after which a function repeats; for $\sin(x)$, $\cos(x)$, $\sec(x)$, and $\csc(x)$ it is $2\pi$, while for $\tan(x)$ and $\cot(x)$ it is $\pi$.

Domain of Sine and Cosine

The set of all real numbers ($R$).

Range of Sine and Cosine

The interval of values between $-1$ and $1$, inclusive, expressed as $[-1, 1]$.

$y = \sin(x)$ Graph Characteristics

A continuous wave-shaped curve that passes through the origin $(0, 0)$ and reaches a maximum value of $1$ and a minimum value of $-1$.