Class 11th Commerce: Complete Math Guide on Radians, Degrees, and Arc Length | Project 45

Angles and radians are fundamental concepts in mathematics, especially when dealing with rotations, circular motion, or even real-life applications like clocks and wheels. In this blog, we will break down complex problems with simple step-by-step explanations and examples, making it easy to understand.

Converting Radians to Degrees (and Vice Versa)

To start, let’s understand how to convert angles from radians to degrees and vice versa:

1. Step 1: Multiply the given radian value by 180π\frac{180}{\pi}π180​ to convert it into degrees.

2. Step 2: Perform the division. Any remainder must be multiplied by 60 to convert it into minutes.

3. Step 3: If there is still a remainder, multiply it again by 60 to convert it into seconds.

4. Step 4: Combine all units (degrees, minutes, seconds) for the final answer.

Example:

A radian value is converted to degrees, minutes, and seconds as follows:

• Final answer: 343° 38′ 11″ (approximated where necessary).

Tip: Degrees are used in linear Cartesian planes, but radians are more useful in circular motion, which is why converting between these units is essential.

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Handling Negative Angles

Negative angles are represented just as they are and affect the direction of rotation. For example, −47°30′-47° 30′−47°30′ converts to radians using the same method:

1. Convert minutes to degrees: 30′=3060=0.5°30′ = \frac{30}{60} = 0.5°30′=6030​=0.5°

2. Combine with degrees: −47°+0.5°=−46.5°-47° + 0.5° = -46.5°−47°+0.5°=−46.5°

3. Convert to radians: −46.5°×π180=−19π72-46.5° \times \frac{\pi}{180} = -\frac{19\pi}{72}−46.5°×180π​=−7219π​

Wheel Rotation Problem

Problem: A wheel completes 360 revolutions in 1 minute. How many radians does it cover in 1 second?

Solution Steps:

1. Calculate revolutions per second: 360÷60=6360 \div 60 = 6360÷60=6

2. Convert each revolution to radians: 1 revolution=2π radians1 \text{ revolution} = 2\pi \text{ radians}1 revolution=2π radians

3. Multiply: 6×2π=12π6 \times 2\pi = 12\pi6×2π=12π radians per second.

Length of Arc

The length of an arc is another important concept derived from the angle subtended at the center of a circle.

Formula:

L=θ⋅rL = \theta \cdot rL=θ⋅r

Where:

• LLL = length of arc

• θ\thetaθ = angle in radians

• rrr = radius of the circle

Example:
A radius of 100 cm with an arc length of 22 cm:

θ=Lr=22100=0.22 radians\theta = \frac{L}{r} = \frac{22}{100} = 0.22 \text{ radians}θ=rL​=10022​=0.22 radians

Convert to degrees:

θdeg=0.22×180π≈12°36′\theta_\text{deg} = 0.22 \times \frac{180}{\pi} \approx 12° 36′θdeg​=0.22×π180​≈12°36′

Practical Application: Minute Hand of a Clock

Problem: A clock's minute hand is 1.5 cm long. How far does its tip move in 40 minutes?

Solution:

1. One hour = 360°, so in 1 minute: 360°÷60=6°360° \div 60 = 6°360°÷60=6°

2. Convert 40 minutes: 6°×40=240°6° \times 40 = 240°6°×40=240°

3. Convert degrees to radians: 240°×π180=4π3 radians240° \times \frac{\pi}{180} = \frac{4\pi}{3} \text{ radians}240°×180π​=34π​ radians

4. Arc length (tip movement): L=θ⋅r=4π3⋅1.5≈6.28 cmL = \theta \cdot r = \frac{4\pi}{3} \cdot 1.5 \approx 6.28 \text{ cm}L=θ⋅r=34π​⋅1.5≈6.28 cm

Key Takeaways for Students

• Always convert degrees to radians when working with circular motion or angles in Class 11.

• Use degrees, minutes, and seconds for exact representation.

• Negative angles indicate direction, positive for anticlockwise rotation.

• Arc length is a practical application of radians in real life.

For examination purposes, practicing NCERT and additional questions is crucial.