Evaluate the angular value of a minute - Class 11 Applied Mathematics

Understanding Angular Units: Degrees and Minutes

In the sexagesimal system, an angle is measured in degrees, minutes, and seconds. A complete rotation is divided into 360 degrees (°). To achieve higher precision, each degree is further subdivided into 60 equal parts called minutes ('). Therefore, one degree contains 60 minutes. The angular value of one minute is defined as 1/60th of a degree.

1° = 60' (minutes) => 1' = (1/60)°

Example 1: Express 1 minute in terms of degrees and subsequently in radians.

Show Step-by-Step Solution

• Step 1: Since 60' = 1°, then 1' = 1/60 degrees.
• Step 2: To convert degrees to radians, multiply by (π/180).
• Step 3: Calculate: (1/60) * (π/180) = π / 10800 radians.

Answer: 1 minute is equal to (1/60)° or π/10800 radians.

Example 2: Convert 15 minutes into degrees.

Show Step-by-Step Solution

• Step 1: Use the relation 1' = (1/60)°.
• Step 2: Multiply the given value by the conversion factor: 15 * (1/60).
• Step 3: Simplify the fraction: 15/60 = 1/4 = 0.25°.

Answer: 15 minutes is equal to 0.25°.

Section A: Multiple Choice Questions (MCQs)

What is the value of 1 minute in terms of degrees?

A. (1/60) degrees B. (1/3600) degrees C. (pi/180) degrees D. (pi/10800) degrees

Show Answer

✔ Correct: A

Since 1 degree = 60 minutes, 1 minute = 1/60 degrees.

What is the value of 1 minute in radians?

A. (pi/180) radians B. (pi/10800) radians C. (pi/648000) radians D. (1/60) radians

Show Answer

✔ Correct: B

1 degree = (pi/180) radians. Therefore, 1 minute = (1/60) * (pi/180) radians = pi/10800 radians.

How many minutes are there in 1 radian?

A. Approx 3438 minutes B. Approx 60 minutes C. Approx 180 minutes D. Approx 3600 minutes

Show Answer

✔ Correct: A

1 radian = (180/pi) degrees approx 57.296 degrees. Since 1 degree = 60 minutes, 57.296 * 60 = 3437.75, which rounds to 3438 minutes.

Section B: Assertion-Reason Questions

Assertion (A): 1 minute is equal to (pi/10800) radians.
Reason (R): There are 60 minutes in a degree and 180 degrees in pi radians.

Directions: Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

Both Assertion and Reason are true and the reason correctly explains the conversion: 1 min = 1/60 deg = (1/60) * (pi/180) rad = pi/10800 rad.

Assertion (A): The angular measure of a second is greater than a minute.
Reason (R): 1 degree = 60 minutes and 1 minute = 60 seconds.

Directions: Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: D

Assertion is false because 1 second (1/3600 deg) is smaller than 1 minute (1/60 deg). Reason is true.

Section C: Short Answer Questions

Convert 25 minutes into degrees.

Show Answer & Solution

Since 60 minutes = 1 degree, then 1 minute = 1/60 degrees. Therefore, 25 minutes = 25/60 degrees = 5/12 degrees, which is approx 0.4167 degrees.

Express 10 minutes in radian measure.

Show Answer & Solution

10 minutes = 10/60 degrees = 1/6 degrees. Converting to radians: (1/6) * (pi/180) = pi/1080 radians.

Section D: Long Answer Questions

Calculate the number of radians in 1 degree, 1 minute, and 1 second.

Show Answer & Solution

Step 1: 1 degree = pi/180 rad = 0.01745 rad. Step 2: 1 minute = (1/60) degree = (1/60) * (pi/180) = pi/10800 rad approx 0.0002908 rad. Step 3: 1 second = (1/60) minute = (1/60) * (pi/10800) = pi/648000 rad approx 0.000004848 rad.

If an angle is given as 2 degrees and 30 minutes, convert this entire angle into radians.

Show Answer & Solution

Step 1: Convert minutes to degrees: 30 minutes = 30/60 = 0.5 degrees. Step 2: Total degrees = 2 + 0.5 = 2.5 degrees. Step 3: Convert to radians: 2.5 * (pi/180) = (5/2) * (pi/180) = 5pi/360 = pi/72 radians.

Section E: Case-Based Questions

Case Study: In surveying and astronomy, precise angular measurement is critical. Angles are often measured in degrees, minutes, and seconds, where 1 degree = 60 minutes and 1 minute = 60 seconds. To perform calculations in calculus or physics, these must be converted into circular measure (radians).

Sub-question 1: If a telescope detects an object moving at a rate of 12 minutes per second, express this movement in degrees per second.

Show Solution

12 minutes per second = 12/60 degrees per second = 0.2 degrees per second.

Sub-question 2: Convert the value 0.2 degrees per second into radians per second.

Show Solution

0.2 * (pi/180) radians per second = 0.2pi/180 = pi/900 radians per second.

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