RBSE Solutions Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.1

Question 1. Find the radian measures corresponding to the following degree measures
(i) 25°
(ii) -47°30'
(iii) 520°
Answer:
(i) To convert degrees to radians, we multiply by \( \frac{\pi}{180^\circ} \).
So, \( 25^\circ = 25 \times \frac{\pi}{180} \) radian.
Simplifying the fraction, we get \( \frac{5\pi}{36} \) radian.

(ii) First, convert 30 minutes to degrees: \( 30' = \frac{30}{60}^\circ = \frac{1}{2}^\circ \).
So, \( -47^\circ 30' = -(47 + \frac{1}{2})^\circ = -\frac{95}{2}^\circ \).
Now convert to radians: \( -\frac{95}{2} \times \frac{\pi}{180} \) radian.
This simplifies to \( -\frac{19\pi}{72} \) radian.

(iii) Convert \( 520^\circ \) to radians:
\( 520^\circ = 520 \times \frac{\pi}{180} \) radian.
Simplifying the fraction, we get \( \frac{26\pi}{9} \) radian. Radian measures are often expressed in terms of pi.
In simple words: To change degrees to radians, multiply the degree value by \( \frac{\pi}{180} \). If there are minutes, change them to a fraction of a degree first.

🎯 Exam Tip: Remember the conversion factor: \( 1^\circ = \frac{\pi}{180} \) radians. Always simplify the fraction to its lowest terms.

Question 2. Find the degree measures corresponding to the following radian measures \( \left( \text{use}\pi =\frac { 22 }{7} \right) \)
(i) \( \frac {11}{16} \)
(ii) -4
(iii) \( \frac {5\pi }{3} \)
Answer:
(i) To convert radians to degrees, we multiply by \( \frac{180^\circ}{\pi} \). We are given to use \( \pi = \frac{22}{7} \).
\( \frac{11}{16} \) radians \( = \frac{11}{16} \times \frac{180^\circ}{\pi} = \frac{11}{16} \times \frac{180^\circ}{\frac{22}{7}} \)
\( = \frac{11}{16} \times \frac{180 \times 7}{22}^\circ = \frac{11 \times 180 \times 7}{16 \times 22}^\circ \)
\( = \frac{1 \times 45 \times 7}{4 \times 2}^\circ = \frac{315}{8}^\circ \)
Now, convert the improper fraction to degrees and minutes:
\( \frac{315}{8}^\circ = 39^\circ + \frac{3}{8}^\circ \)
To convert \( \frac{3}{8}^\circ \) to minutes, multiply by 60:
\( \frac{3}{8} \times 60' = \frac{180}{8}' = \frac{45}{2}' = 22.5' \)
So, \( 39^\circ + 22.5' = 39^\circ 22' 30'' \). We split 0.5 minutes into 30 seconds (0.5 * 60 = 30).

(ii) Convert -4 radians to degrees, using \( \pi = \frac{22}{7} \).
\( -4 \) radians \( = -4 \times \frac{180^\circ}{\pi} = -4 \times \frac{180^\circ}{\frac{22}{7}} \)
\( = -4 \times \frac{180 \times 7}{22}^\circ = -\frac{4 \times 180 \times 7}{22}^\circ \)
\( = -\frac{2 \times 180 \times 7}{11}^\circ = -\frac{2520}{11}^\circ \)
Now, convert the improper fraction to degrees, minutes, and seconds:
\( -\frac{2520}{11}^\circ = -(229^\circ + \frac{1}{11}^\circ) \)
To convert \( \frac{1}{11}^\circ \) to minutes, multiply by 60:
\( \frac{1}{11} \times 60' = \frac{60}{11}' = 5' + \frac{5}{11}' \)
To convert \( \frac{5}{11}' \) to seconds, multiply by 60:
\( \frac{5}{11} \times 60'' = \frac{300}{11}'' \approx 27.27'' \)
So, \( -4 \) radians \( \approx -(229^\circ 5' 27.3'') \). We round the seconds to one decimal place.

(iii) Convert \( \frac{5\pi}{3} \) radians to degrees.
\( \frac{5\pi}{3} \) radians \( = \frac{5\pi}{3} \times \frac{180^\circ}{\pi} \)
The \( \pi \) terms cancel out, simplifying the calculation.
\( = \frac{5 \times 180}{3}^\circ = 5 \times 60^\circ = 300^\circ \).
In simple words: To change radians to degrees, multiply the radian value by \( \frac{180}{\pi} \). Remember that 1 degree has 60 minutes and 1 minute has 60 seconds, which helps in converting decimal degrees.

🎯 Exam Tip: Pay close attention to unit conversions for degrees, minutes, and seconds. If \( \pi \) is present in the radian measure, it often cancels out in the conversion. If not, use the given approximation for \( \pi \), usually \( \frac{22}{7} \) or 3.14.

Question 3. A wheel makes 360 revolutions in 1 minute than how many radians does it turn in one second ?
Answer:
The wheel makes 360 revolutions in 1 minute.
We know that 1 minute has 60 seconds.
So, in 60 seconds, the wheel makes 360 revolutions.
To find revolutions in one second, divide total revolutions by total seconds:
Revolutions in 1 second \( = \frac{360 \text{ revolutions}}{60 \text{ seconds}} = 6 \text{ revolutions/second} \).
One full revolution is equal to \( 2\pi \) radians.
So, the angle made in 6 revolutions \( = 6 \times 2\pi \) radians \( = 12\pi \) radians.
Therefore, the wheel turns \( 12\pi \) radians in one second. This shows the angular speed of the wheel.
In simple words: First, find how many times the wheel spins in one second. Then, since one full spin is \( 2\pi \) radians, multiply that by the number of spins per second to get the total radians.

🎯 Exam Tip: Always convert time units to match (minutes to seconds) before calculating rates. Remember that one revolution is equivalent to \( 2\pi \) radians, or \( 360^\circ \).

Question 4. Find the degree measure of the angle sub-tended at the centre of a circle of radius 100 cm by an arc of length 22 cm \( \left(\text{use}\pi =\frac {22}{7} \right) \)
Answer:
Given: Radius of circle \( (r) = 100 \) cm.
Length of arc \( (l) = 22 \) cm.
The formula for the angle \( (\theta) \) subtended at the center by an arc is \( \theta = \frac{l}{r} \), where \( \theta \) is in radians.
\( \theta = \frac{22 \text{ cm}}{100 \text{ cm}} = \frac{22}{100} \) radians.
Now, convert this angle from radians to degrees, using \( \pi = \frac{22}{7} \).
\( \theta \) in degrees \( = \frac{22}{100} \times \frac{180^\circ}{\pi} = \frac{22}{100} \times \frac{180^\circ}{\frac{22}{7}} \)
\( = \frac{22}{100} \times \frac{180 \times 7}{22}^\circ = \frac{180 \times 7}{100}^\circ \)
\( = \frac{18 \times 7}{10}^\circ = \frac{126}{10}^\circ = 12.6^\circ \).
To express this in degrees and minutes: \( 12.6^\circ = 12^\circ + 0.6^\circ \).
Convert \( 0.6^\circ \) to minutes: \( 0.6 \times 60' = 36' \).
So, the angle subtended at the center is \( 12^\circ 36' \). It's important to specify the units correctly.
In simple words: First, find the angle in radians by dividing the arc length by the radius. Then, change this radian value into degrees and minutes using the conversion factor, remembering that 1 degree has 60 minutes.

🎯 Exam Tip: Always ensure the angle calculated using \( \theta = \frac{l}{r} \) is understood to be in radians. Don't forget to convert it to degrees, minutes, and seconds if the question asks for degree measure.

Question 5. In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of corresponding chord.
Answer:
Given: Diameter of the circle \( = 40 \) cm.
So, the radius of the circle \( (r) = \frac{40}{2} = 20 \) cm.
Length of the chord \( AB = 20 \) cm.
Consider the triangle formed by the two radii and the chord, say \( \triangle OAB \), where O is the center.
We have \( OA = r = 20 \) cm, \( OB = r = 20 \) cm, and \( AB = 20 \) cm.
Since all three sides of \( \triangle OAB \) are 20 cm, it is an equilateral triangle.
In an equilateral triangle, all angles are \( 60^\circ \).
So, the angle subtended by the chord at the center, \( \theta = \angle AOB = 60^\circ \).
Convert this angle to radians: \( \theta = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \) radians.
Now, use the arc length formula: \( l = r\theta \).
Length of minor arc \( = 20 \times \frac{\pi}{3} \) cm \( = \frac{20\pi}{3} \) cm.
Using \( \pi \approx \frac{22}{7} \):
Arc length \( = \frac{20}{3} \times \frac{22}{7} = \frac{440}{21} \approx 20.95 \) cm.
The length of the minor arc is approximately 20.95 cm. This links geometry with trigonometry.
In simple words: First, use the diameter to find the radius. Then, since the chord length is also equal to the radius, the triangle formed by the radii and the chord is equilateral, making the central angle \( 60^\circ \). Convert this angle to radians and use the formula \( \text{arc length} = \text{radius} \times \text{angle in radians} \).

🎯 Exam Tip: When a chord length equals the radius, it always forms an equilateral triangle with the center, meaning the central angle is \( 60^\circ \). This is a common shortcut in geometry problems.

Question 6. If in two circles, arcs of the same length subtend angles of 60° and 75° at the centre, find the ratio of their radii.
Answer:
Let the radii of the two circles be \( r_1 \) and \( r_2 \).
Let the common arc length be \( l \).
For the first circle:
Angle subtended \( \theta_1 = 60^\circ \). Convert to radians: \( \theta_1 = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \) radians.
Using the formula \( l = r_1 \theta_1 \), we have \( l = r_1 \left(\frac{\pi}{3}\right) \). So, \( r_1 = \frac{3l}{\pi} \).

For the second circle:
Angle subtended \( \theta_2 = 75^\circ \). Convert to radians: \( \theta_2 = 75 \times \frac{\pi}{180} = \frac{5\pi}{12} \) radians.
Using the formula \( l = r_2 \theta_2 \), we have \( l = r_2 \left(\frac{5\pi}{12}\right) \). So, \( r_2 = \frac{12l}{5\pi} \).

Now, find the ratio of their radii, \( r_1 : r_2 \):
\( \frac{r_1}{r_2} = \frac{\frac{3l}{\pi}}{\frac{12l}{5\pi}} = \frac{3l}{\pi} \times \frac{5\pi}{12l} \)
The \( l \) and \( \pi \) terms cancel out, simplifying the ratio.
\( = \frac{3 \times 5}{12} = \frac{15}{12} = \frac{5}{4} \).
Therefore, the ratio of the radii is \( r_1 : r_2 = 5:4 \). This illustrates the inverse relationship between angle and radius for a fixed arc length.
In simple words: Since both circles have the same arc length, we can set up an equation for each circle using \( \text{arc length} = \text{radius} \times \text{angle in radians} \). Convert angles to radians first. Then, divide the two radius equations to find their ratio.

🎯 Exam Tip: Always convert angles to radians when using the arc length formula \( l = r\theta \). Remember that if the arc length is constant, the radius and the central angle are inversely proportional.

Question 7. Find the angle in radian through which a pendulum swings, if its length is 75 cm and the tip describes an arc of length
(i) 10 cm
(ii) 21 cm
Answer:
The length of the pendulum acts as the radius \( (r) \) of the circular arc its tip describes.
The arc length described by the tip is \( (l) \).
The angle \( (\theta) \) in radians is given by the formula \( \theta = \frac{l}{r} \).

(i) Length of pendulum \( (r) = 75 \) cm.
Arc length \( (l) = 10 \) cm.
\( \theta = \frac{10}{75} \) radians.
Simplifying the fraction, we get \( \theta = \frac{2}{15} \) radians.

(ii) Length of pendulum \( (r) = 75 \) cm.
Arc length \( (l) = 21 \) cm.
\( \theta = \frac{21}{75} \) radians.
Simplifying the fraction by dividing both by 3, we get \( \theta = \frac{7}{25} \) radians.
The angle through which the pendulum swings is directly proportional to the arc length.
In simple words: A pendulum's length is like the radius of a circle, and the path its tip makes is the arc. Just divide the arc length by the pendulum's length to get the angle it swings through, measured in radians.

🎯 Exam Tip: Always ensure the units of arc length and pendulum length are the same before performing the division. The resulting angle from \( \theta = \frac{l}{r} \) is always in radians.