Trigonometry: Concept of Measurement of Angle

In elementary geometry, an angle is a static shape formed by two intersecting line segments. In trigonometry, an angle is defined dynamically as the measure of rotation of a ray:

• Vertex (শীর্ষবিন্দু, $O$): The fixed center of rotation.
• Initial Arm (প্রারম্ভিক বাহু, $OX$): The fixed starting position of the revolving ray.
• Terminal Arm (প্রান্তিক বাহু, $OP$): The final position of the ray after rotation.
• Magnitude of the Angle: The total amount of rotation executed by the ray from $OX$ to $OP$.

The sign of a trigonometric angle is determined entirely by the direction of rotation of the revolving ray:

Unlike geometric angles, which are bounded between $0^\circ$ and $360^\circ$, a trigonometric ray can rotate indefinitely through multiple full revolutions:

• 1 complete counterclockwise revolution $= +360^\circ$
• 2 complete counterclockwise revolutions $= +720^\circ$
• $n$ complete counterclockwise revolutions plus an angle $ heta$: $\mathbf{ ext{Angle} = n imes 360^\circ + heta}$
• Example: If a ray makes 2 complete clockwise rotations and then rotates another $45^\circ$ clockwise, the trigonometric angle is $-(2 imes 360^\circ + 45^\circ) = -765^\circ$.

In the Sexagesimal System, the fundamental reference unit is the Right Angle (সমকোণ):

• 1 Right Angle (১ সমকোণ) $= 90^\circ$ (90 degrees).
• 1 Degree ($1^\circ$) $= 60'$ (60 minutes of arc).
• 1 Minute ($1'$) $= 60''$ (60 seconds of arc).
• Therefore: $1^\circ = 60' = 3600''$.

In scientific analysis and higher mathematics, angles are measured in Radians (রেডিয়ান):

Theorem: The radian is a constant angle independent of the radius of the circle.

Proof:

1. Consider a circle with center $O$ and radius $r$. Let arc $AB$ have length equal to the radius: $ ext{arc } AB = r$.
2. By the definition of a radian, the central angle subtended by this arc is 1 radian: $$ngle AOB = 1 ext{ radian} = 1^c$$
3. In any circle, the angles subtended at the center by two arcs are directly proportional to the lengths of the arcs: $$rac{ngle AOB}{ ext{Straight Angle}} = rac{ ext{length of arc } AB}{ ext{length of semicircle arc}}$$
4. We know that: $$ ext{length of arc } AB = r$$ $$ ext{length of semicircular arc} = rac{1}{2} imes (2\pi r) = \pi r$$ $$ ext{Straight Angle} = 2 ext{ right angles} = 180^\circ$$
5. Substituting these values into the proportion: $$rac{1 ext{ radian}}{2 ext{ right angles}} = rac{r}{\pi r} = rac{1}{\pi}$$ $$\mathbf{1 ext{ radian} = rac{2 ext{ right angles}}{\pi} = rac{180^\circ}{\pi}}$$
6. Since 2 right angles is a fixed constant and $\pi$ is an absolute mathematical constant (the ratio of circumference to diameter), their quotient $rac{2 ext{ right angles}}{\pi}$ is an unchanging constant.
7. Therefore, the Radian is a constant angle. (Hence Proved)

From the relation $1 ext{ radian} = rac{180^\circ}{\pi}$, cross-multiplying yields the cornerstone identity of trigonometry:

To convert an angle expressed in degrees, minutes, and seconds ($D^\circ M' S''$) into circular measure:

1. Convert seconds into minutes: $M' + rac{S}{60}' = M_{ ext{total}}'$.
2. Convert minutes into degrees: $D^\circ + rac{M_{ ext{total}}}{60}^\circ = D_{ ext{total}}^\circ$.
3. Multiply the total degree value by $rac{\pi}{180}$: $$\mathbf{ heta_{ ext{radians}} = D_{ ext{total}} imes rac{\pi}{180}}$$

To convert an angle from radians to degrees, minutes, and seconds:

1. Multiply by $rac{180^\circ}{\pi}$ (substituting $\pi = rac{22}{7}$ if numerical evaluation is required).
2. The whole number part of the quotient gives the Degrees ($^\circ$).
3. Multiply the fractional degree remainder by $60$. The whole number part of the result gives the Minutes ($'$).
4. Multiply the fractional minute remainder by $60$. The resulting integer/rounded value gives the Seconds ($''$).

In a circle of radius $r$, consider an arc of length $s$ that subtends a central angle $ heta$ (in radians):

By direct proportion with the complete circle area $\pi r^2$ and full revolution $2\pi$ radians: $$rac{ ext{Area of Sector } A}{ ext{Total Area } \pi r^2} = rac{ heta}{2\pi} \implies \mathbf{A = rac{1}{2} r^2 heta = rac{1}{2} r s}$$

High-frequency board problems require understanding angular speeds of clock hands:

• Minute Hand: Rotates $360^\circ$ ($2\pi ext{ rad}$) in 60 minutes. $$ ext{Angular Speed} = rac{360^\circ}{60} = \mathbf{6^\circ ext{ per minute}} = \mathbf{rac{\pi}{30} ext{ rad/min}}$$ Note: The minute hand rotates clockwise, so its trigonometric angle is negative!
• Hour Hand: Rotates $360^\circ$ in 12 hours (720 minutes). $$ ext{Angular Speed} = rac{360^\circ}{720} = \mathbf{0.5^\circ ext{ per minute}} = \mathbf{rac{1}{2}^\circ ext{ per minute}}$$
• Interior Angle of a Regular Polygon with n sides: $$ ext{Each Interior Angle} = \mathbf{rac{(n - 2) imes 180^\circ}{n} = rac{(n - 2)\pi}{n} ext{ radians}}$$