Degrees to Radians Conversion & Arcs

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✖️ 1. Understanding radians as a natural geometric measure of rotation

📐 Radians: The Natural Angle

A radian is the angle where the arc length equals the radius.
One full circle contains exactly $2\pi$ radians.
Radians measure rotation by comparing arc length to radius.
The formula is $\theta = \frac{s}{r}$ where $s$ is arc length and $r$ is radius.
No units needed because radians are a pure ratio.

If a circle has radius 5 cm and arc length 5 cm, the angle is exactly 1 radian.

💡 Think: Wrap the radius around the circle—that's 1 radian!

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1. Understanding radians as a natural geometric measure of rotation

Understanding Radians as a Natural Geometric Measure

A radian is the angle subtended at the center of a circle when the arc length equals the radius. Unlike degrees, which divide a circle arbitrarily into 360 parts, radians emerge directly from the circle's geometry.

Intuition: If you wrap the radius along the circle's edge, the angle formed at the center is exactly 1 radian. This makes radians dimensionless (length divided by length) and inherently tied to circular motion.

Core Rules:

One full rotation equals $2\pi$ radians (since circumference $= 2\pi r$ )
Half rotation (straight angle) = $\pi$ radians
Quarter rotation (right angle) = $\frac{\pi}{2}$ radians
Radians measure the ratio $\theta = \frac{s}{r}$ where $s$ is arc length and $r$ is radius

This definition makes calculus and physics formulas simpler, as derivatives of trigonometric functions work naturally only in radians.

Example: An arc of length 5 units on a circle of radius 5 units subtends an angle of $\theta = \frac{5}{5} = 1$ radian.

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A circle has a radius of $4$ units. An arc on this circle has a length of $12$ units. Calculate the angle subtended by this arc at the center, in radians.

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✖️ 2. Converting between degrees and radians

🔄 Degree-Radian Conversion

The key fact: $\pi$ radians = 180 degrees.
To convert degrees to radians: multiply by $\frac{\pi}{180}$ .
To convert radians to degrees: multiply by $\frac{180}{\pi}$ .
Common angles: 90 degrees = $\frac{\pi}{2}$ rad, 60 degrees = $\frac{\pi}{3}$ rad, 45 degrees = $\frac{\pi}{4}$ rad.

Convert 120 degrees to radians: $120 \times \frac{\pi}{180} = \frac{2\pi}{3}$ rad.

💡 Remember: $\pi$ is the bridge between both systems!

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2. Converting between degrees and radians

Converting Between Degrees and Radians

The conversion between degrees and radians relies on the fundamental equivalence: $\pi$ radians $= 180^\circ$ . This relationship allows bidirectional conversion using multiplication by appropriate conversion factors.

Intuition: Since a full circle is both $360^\circ$ and $2\pi$ radians, half that amount gives the key ratio. Every degree corresponds to $\frac{\pi}{180}$ radians, and every radian corresponds to $\frac{180}{\pi}$ degrees.

Core Rules:

Degrees to radians: Multiply by $\frac{\pi}{180}$ , so $\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}$
Radians to degrees: Multiply by $\frac{180}{\pi}$ , so $\theta_{\text{deg}} = \theta_{\text{rad}} \cdot \frac{180}{\pi}$
Common angles: $90^\circ = \frac{\pi}{2}$ , $45^\circ = \frac{\pi}{4}$ , $60^\circ = \frac{\pi}{3}$

Memorizing common conversions accelerates problem-solving in trigonometry and calculus.

Example: Convert $120^\circ$ to radians: $120 \cdot \frac{\pi}{180} = \frac{2\pi}{3}$ radians.

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Convert $30^\circ$ to radians.

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✖️ 3. Calculating arc length and sector area in radians

📏 Arc Length and Sector Area

Arc length formula: $s = r\theta$ where $\theta$ is in radians.
Sector area formula: $A = \frac{1}{2}r^2\theta$ where $\theta$ is in radians.
These formulas ONLY work when angle is in radians.
Larger angle means longer arc and bigger sector area.

Circle with radius 4 m and angle $\frac{\pi}{3}$ rad: arc length $s = 4 \times \frac{\pi}{3} = \frac{4\pi}{3}$ m, area $A = \frac{1}{2}(4)^2 \times \frac{\pi}{3} = \frac{8\pi}{3}$ square m.

💡 Visual: Arc length is radius times angle—simple multiplication!

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3. Calculating arc length and sector area in radians

Calculating Arc Length and Sector Area

When angles are measured in radians, arc length and sector area formulas simplify to direct proportional relationships. The arc length $s$ is the product of radius and angle: $s = r\theta$ . The sector area $A$ is half the product of radius squared and angle: $A = \frac{1}{2}r^2\theta$ .

Intuition: Since $\theta$ in radians already represents the arc-to-radius ratio, multiplying by $r$ recovers the arc length. The sector area formula mirrors the triangle area formula $\frac{1}{2}bh$ , treating the arc as the "base."

Core Rules:

Arc length: $s = r\theta$ (where $\theta$ is in radians)
Sector area: $A = \frac{1}{2}r^2\theta$ (where $\theta$ is in radians)
These formulas fail if $\theta$ is in degrees; conversion is mandatory
For a full circle: $s = 2\pi r$ and $A = \pi r^2$

These formulas are foundational in physics for rotational motion and engineering applications.

Example: A sector with radius 4 and angle $\frac{\pi}{3}$ has arc length $s = 4 \cdot \frac{\pi}{3} = \frac{4\pi}{3}$ and area $A = \frac{1}{2}(4)^2 \cdot \frac{\pi}{3} = \frac{8\pi}{3}$ .

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A circle has a radius of 5 units. Find the arc length intercepted by a central angle of 2 radians.

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✖️ 4. Identifying and calculating coterminal angles

🔁 Coterminal Angles

Coterminal angles share the same terminal side after rotation.
In degrees: add or subtract multiples of 360 degrees.
In radians: add or subtract multiples of $2\pi$ radians.
Formula for degrees: $\theta + 360k$ where $k$ is any integer.
Formula for radians: $\theta + 2\pi k$ where $k$ is any integer.

Coterminal with 45 degrees: 405 degrees (add 360) or -315 degrees (subtract 360). Coterminal with $\frac{\pi}{4}$ rad: $\frac{9\pi}{4}$ rad (add $2\pi$ ).

💡 Think: Full rotations don't change where you end up!

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4. Identifying and calculating coterminal angles

Identifying and Calculating Coterminal Angles

Coterminal angles are angles that share the same terminal side when drawn in standard position, differing by full rotations. In degrees, add or subtract multiples of $360^\circ$ ; in radians, add or subtract multiples of $2\pi$ .

Intuition: Rotating an additional full circle (or multiple circles) returns you to the same position, so angles separated by complete rotations are geometrically identical.

Core Rules:

In degrees: $\theta_{\text{coterm}} = \theta + 360^\circ k$ where $k \in \mathbb{Z}$
In radians: $\theta_{\text{coterm}} = \theta + 2\pi k$ where $k \in \mathbb{Z}$
Positive coterminal: Use $k > 0$ (add rotations)
Negative coterminal: Use $k < 0$ (subtract rotations)
To find the principal angle (smallest positive), reduce modulo $360^\circ$ or $2\pi$

Coterminal angles have identical trigonometric function values.

Example: Coterminal with $\frac{\pi}{4}$ : adding $2\pi$ gives $\frac{\pi}{4} + 2\pi = \frac{9\pi}{4}$ ; subtracting $2\pi$ gives $\frac{\pi}{4} - 2\pi = -\frac{7\pi}{4}$ .

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Find the smallest positive coterminal angle (in degrees) for $-60^\circ$ . Enter only the number.

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✖️ 5. Applications in angular velocity and rotational kinematics

⚙️ Angular Velocity and Rotation

Angular velocity $\omega$ measures how fast something rotates in radians per second.
Formula: $\omega = \frac{\theta}{t}$ where $\theta$ is angle in radians and $t$ is time.
Linear velocity relates to angular velocity: $v = r\omega$ .
Radians are required for physics formulas to work correctly.
Higher $\omega$ means faster rotation.

A wheel rotates through $4\pi$ rad in 2 seconds: $\omega = \frac{4\pi}{2} = 2\pi$ rad per second. If radius is 0.5 m, linear velocity $v = 0.5 \times 2\pi = \pi$ m per second.

💡 Remember: Radians make rotation math work seamlessly with distance!

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5. Applications in angular velocity and rotational kinematics

Applications: Angular Velocity and Rotational Kinematics

Angular velocity $\omega$ measures the rate of angular displacement, defined as $\omega = \frac{\Delta \theta}{\Delta t}$ where $\theta$ is in radians and $t$ is time. Radians are mandatory here because they make the relationship between linear and angular quantities dimensionally consistent.

Intuition: Just as linear velocity measures distance per time, angular velocity measures angle per time. Using radians ensures that $v = r\omega$ (linear velocity equals radius times angular velocity) works without conversion factors.

Core Rules:

Angular velocity: $\omega = \frac{d\theta}{dt}$ (radians per second)
Linear velocity relation: $v = r\omega$ (only valid when $\omega$ is in rad/s)
Angular acceleration: $\alpha = \frac{d\omega}{dt}$ (radians per second squared)
Rotational kinematic equations mirror linear ones: $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$

These formulas are essential in mechanics, robotics, and orbital dynamics.

Example: A wheel rotating at $\omega = 5$ rad/s with radius 0.2 m has linear edge velocity $v = 0.2 \cdot 5 = 1$ m/s.

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A gear has a radius of $0.5$ m and rotates with an angular velocity of $6$ rad/s. Calculate its linear edge velocity in m/s.

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Degrees and radians

✖️ 1. Understanding radians as a natural geometric measure of rotation

📐 Radians: The Natural Angle

1. Understanding radians as a natural geometric measure of rotation

Understanding Radians as a Natural Geometric Measure

✖️ 2. Converting between degrees and radians

🔄 Degree-Radian Conversion

2. Converting between degrees and radians

Converting Between Degrees and Radians

✖️ 3. Calculating arc length and sector area in radians

📏 Arc Length and Sector Area

3. Calculating arc length and sector area in radians

Calculating Arc Length and Sector Area

✖️ 4. Identifying and calculating coterminal angles

🔁 Coterminal Angles

4. Identifying and calculating coterminal angles

Identifying and Calculating Coterminal Angles

✖️ 5. Applications in angular velocity and rotational kinematics

⚙️ Angular Velocity and Rotation

5. Applications in angular velocity and rotational kinematics

Applications: Angular Velocity and Rotational Kinematics

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