GSEB Class 11 Maths Solutions Chapter 3 Trigonometric Functions Exercise 3.1

Question 1. Find the radian measures corresponding to the following degree measures?
(i) \( 25^\circ \)
(ii) \( 47^\circ 30' \)
(iii) \( 240^\circ \)
(iv) \( 520^\circ \)
Answer:
(i) For \( 25^\circ \):
We know that \( 180^\circ \) is equal to \( \pi \) radians. So, to convert \( 25^\circ \) to radians, we multiply it by \( \frac{\pi}{180} \).
\( \implies 25^\circ = 25 \times \frac{\pi}{180} \) radians
\( \implies = \frac{5\pi}{36} \) radians.
(ii) For \( 47^\circ 30' \):
First, we convert 30 minutes to degrees: \( 30' = \frac{30}{60}^\circ = \frac{1}{2}^\circ \). So, \( 47^\circ 30' = \left(47 + \frac{1}{2}\right)^\circ = \left(\frac{94+1}{2}\right)^\circ = \frac{95}{2}^\circ \). We use the conversion \( 180^\circ = \pi \) radians. Therefore, \( \frac{95}{2}^\circ = \frac{95}{2} \times \frac{\pi}{180} \) radians. This simplifies to \( \frac{19\pi}{72} \) radians.
(iii) For \( 240^\circ \):
Knowing that \( 180^\circ = \pi \) radians, we convert \( 240^\circ \) by multiplying it by \( \frac{\pi}{180} \).
\( \implies 240^\circ = 240 \times \frac{\pi}{180} \) radians
\( \implies = \frac{4\pi}{3} \) radians.
(iv) For \( 520^\circ \):
Again, using the conversion \( 180^\circ = \pi \) radians, we convert \( 520^\circ \) to radians by multiplying by \( \frac{\pi}{180} \).
\( \implies 520^\circ = 520 \times \frac{\pi}{180} \) radians
\( \implies = \frac{26\pi}{9} \) radians.
In simple words: To change degrees to radians, you multiply the degree value by \( \frac{\pi}{180} \). Remember to convert minutes to degrees first if they are present.

Exam Tip: Always make sure to convert any minutes or seconds into decimal degrees before applying the degree-to-radian conversion formula to avoid calculation errors.

Question 2. Find the degree measures corresponding to the following radian measures. [Use \( \pi = \frac{22}{7} \)]
(i) \( \frac{11}{16} \)
(ii) \( -4 \)
(iii) \( \frac{5\pi}{3} \)
(iv) \( \frac{7\pi}{6} \)
Answer:
(i) For \( \frac{11}{16} \) radians:
We know that \( \pi \) radians is equal to \( 180^\circ \). Given \( \pi = \frac{22}{7} \), we use the conversion factor \( \frac{180^\circ}{\pi} \).
\( \implies \frac{11}{16} \) radians \( = \frac{11}{16} \times \frac{180^\circ}{\pi} = \frac{11}{16} \times \frac{180^\circ}{\frac{22}{7}} = \frac{11}{16} \times \frac{180 \times 7}{22}^\circ \)
\( \implies = \frac{1 \times 90 \times 7}{16 \times 1}^\circ = \frac{315}{8}^\circ \)
\( \implies \frac{315}{8}^\circ = 39 \frac{3}{8}^\circ = 39^\circ + \frac{3}{8} \times 60' = 39^\circ + \frac{180}{8}' = 39^\circ + \frac{45}{2}' = 39^\circ + 22.5' \)
\( \implies 22.5' = 22' + 0.5' = 22' + 0.5 \times 60'' = 22' + 30'' \) Thus, \( \frac{11}{16} \) radians \( = 39^\circ 22' 30'' \).
(ii) For \( -4 \) radians:
To convert \( -4 \) radians to degrees, we multiply by \( \frac{180^\circ}{\pi} \). Using \( \pi = \frac{22}{7} \), the calculation becomes:
\( \implies -4 \times \frac{180^\circ}{\frac{22}{7}} = -4 \times \frac{180 \times 7}{22}^\circ = -\frac{2520}{11}^\circ \)
\( \implies -\frac{2520}{11}^\circ = -229 \frac{1}{11}^\circ \) Divide \( 2520 \) by \( 11 \): 11)2520(229    -22    ----     32     -22     ----      100      -99      ----        1 Now, convert the fractional part \( \frac{1}{11}^\circ \) to minutes:
\( \implies \frac{1}{11}^\circ = \frac{1}{11} \times 60' = \frac{60}{11}' = 5 \frac{5}{11}' \) Divide \( 60 \) by \( 11 \): 11)60(5 minutes    -55    ----     5 minutes Finally, convert the fractional part \( \frac{5}{11}' \) to seconds:
\( \implies \frac{5}{11}' = \frac{5}{11} \times 60'' = \frac{300}{11}'' \approx 27'' \) Divide \( 300 \) by \( 11 \): 11)300(27 seconds    -22    ----     80     -77     ----      3 Therefore, \( -4 \) radians is approximately \( -229^\circ 5' 27'' \).
(iii) For \( \frac{5\pi}{3} \) radians:
To convert \( \frac{5\pi}{3} \) radians to degrees, we use the conversion factor \( \frac{180^\circ}{\pi} \).
\( \implies \frac{5\pi}{3} \) radians \( = \frac{5\pi}{3} \times \frac{180^\circ}{\pi} \)
\( \implies = 5 \times 60^\circ = 300^\circ \).
(iv) For \( \frac{7\pi}{6} \) radians:
For \( \frac{7\pi}{6} \) radians, we multiply by \( \frac{180^\circ}{\pi} \) to convert to degrees.
\( \implies \frac{7\pi}{6} \) radians \( = \frac{7\pi}{6} \times \frac{180^\circ}{\pi} \)
\( \implies = 7 \times 30^\circ = 210^\circ \).
In simple words: To change radians to degrees, multiply the radian value by \( \frac{180}{\pi} \). Remember to substitute the given value for \( \pi \), and then break down any decimal degrees into minutes and seconds.

Exam Tip: Be very careful when converting decimal degrees to minutes and seconds. Multiply the fractional part by 60 for minutes, then multiply the fractional part of minutes by 60 for seconds. Don't forget the negative sign for negative radian values.

Question 3. A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?
Answer: We know that one full revolution is equivalent to \( 2\pi \) radians. If the wheel makes \( 360 \) revolutions in one minute, then the total angle rotated in one minute is \( 360 \times 2\pi \) radians. Since one minute equals \( 60 \) seconds, the angle turned in one second will be this total angle divided by \( 60 \).
\( \implies \) Angle for one revolution \( = 2\pi \) radians.
\( \implies \) Angle for \( 360 \) revolutions \( = 360 \times 2\pi \) radians.
\( \implies \) This angle is turned in \( 1 \) minute \( = 60 \) seconds.
\( \implies \) Therefore, angle turned in \( 1 \) second \( = \frac{360 \times 2\pi}{60} = 6 \times 2\pi = 12\pi \) radians.
In simple words: A full spin is \( 2\pi \) radians. If a wheel spins \( 360 \) times in a minute, it covers \( 360 \times 2\pi \) radians. To find how much it spins in one second, divide that total by \( 60 \).

Exam Tip: Always pay attention to the units requested in the question (radians, degrees, seconds, minutes) and ensure all conversions are completed accurately.

Question 4. Find the degree measure of the angle subtended at the centre of a circle of radius 100 cm by an arc of length 22 cm. (Use \( \pi = \frac{22}{7} \))
Answer: We know the formula \( l = r\theta \), where \( l \) is the arc length, \( r \) is the radius, and \( \theta \) is the angle in radians. We are given the radius \( r = 100 \) cm and the arc length \( l = 22 \) cm. So, we can find the angle in radians:
\( \implies \theta = \frac{l}{r} = \frac{22}{100} = 0.22 \) radians. To convert this angle from radians to degrees, we multiply by \( \frac{180^\circ}{\pi} \). Using \( \pi = \frac{22}{7} \), we have:
\( \implies \theta = 0.22 \times \frac{180^\circ}{\pi} = 0.22 \times \frac{180^\circ}{\frac{22}{7}} = \frac{22}{100} \times \frac{180^\circ \times 7}{22} \)
\( \implies = \frac{180^\circ \times 7}{100} = \frac{1260^\circ}{100} = 12.6^\circ \). To express this in degrees and minutes, we separate the whole number and fractional parts: \( 12.6^\circ = 12^\circ + 0.6^\circ \). We convert \( 0.6^\circ \) to minutes:
\( \implies 0.6^\circ = 0.6 \times 60' = 36' \). Therefore, the angle is \( 12^\circ 36' \). Division of \( 360 \) by \( 10 \): 10)360(36 minutes    -360    ----     X
In simple words: The angle in radians is arc length divided by radius. Convert this radian angle to degrees by multiplying by \( \frac{180}{\pi} \). Then, turn any decimal part of the degree into minutes.

Exam Tip: Remember the basic formula \( \theta = \frac{l}{r} \) where \( \theta \) is always in radians. Convert to degrees only as the final step if the question asks for it.

Question 5. In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of the minor arc corresponding to the chord?
Answer: Given that the diameter of the circle is \( 40 \) cm, the radius \( r \) will be half of this, so \( r = 20 \) cm. The length of the chord is also given as \( 20 \) cm. Since the radius and the chord length are both \( 20 \) cm, the triangle formed by the center and the endpoints of the chord (let's call it \( \triangle OAB \)) is an equilateral triangle.
\( \implies \) In an equilateral triangle, all angles are \( 60^\circ \). Therefore, the angle subtended at the center, \( \theta \), is \( 60^\circ \). We need to convert this angle to radians before calculating the arc length.
\( \implies \theta = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \) radians. Now, we use the formula for arc length: \( l = r\theta \). Substituting the values,
\( \implies l = 20 \times \frac{\pi}{3} = \frac{20\pi}{3} \) cm. This is the length of the minor arc.
In simple words: The diameter tells you the radius. If the chord length equals the radius, the triangle at the center is equilateral, so the angle is \( 60^\circ \). Convert this to radians and use \( l = r\theta \) to find the arc length.

Exam Tip: Recognize special triangles like equilateral triangles based on the given dimensions. This will directly help you find the central angle needed for arc length calculations.

Question 6. If, in two circles, arcs of the same length subtend angles of \( 60^\circ \) and \( 75^\circ \) at their centres, find the ratio of their radii?
Answer: Let \( l \) be the common arc length for both circles. For the first circle, the angle subtended is \( \theta_1 = 60^\circ \), and for the second circle, it is \( \theta_2 = 75^\circ \). We first convert these angles to radians, as the arc length formula uses radians.
\( \implies \theta_1 = 60^\circ = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \) radians.
\( \implies \theta_2 = 75^\circ = 75 \times \frac{\pi}{180} = \frac{5\pi}{12} \) radians. We use the formula \( l = r\theta \). So, for the first circle, \( l = r_1 \theta_1 = r_1 \left(\frac{\pi}{3}\right) \). For the second circle, \( l = r_2 \theta_2 = r_2 \left(\frac{5\pi}{12}\right) \). Since the arc lengths are equal, we can set these two expressions equal to each other:
\( \implies r_1 \left(\frac{\pi}{3}\right) = r_2 \left(\frac{5\pi}{12}\right) \) We can cancel \( \pi \) from both sides.
\( \implies \frac{r_1}{3} = \frac{5r_2}{12} \) Now, multiply both sides by \( 12 \) to clear the denominators:
\( \implies 12 \times \frac{r_1}{3} = 12 \times \frac{5r_2}{12} \)
\( \implies 4r_1 = 5r_2 \) To find the ratio \( r_1:r_2 \), we rearrange the equation:
\( \implies \frac{r_1}{r_2} = \frac{5}{4} \) Thus, the ratio of their radii \( r_1:r_2 \) is \( 5:4 \).
In simple words: Convert both angles to radians. Since the arc lengths are the same, set \( r_1\theta_1 \) equal to \( r_2\theta_2 \). Then, solve the equation to find the ratio of the radii.

Exam Tip: Remember to always convert angles to radians when working with arc length or area of sector formulas, as these formulas are derived using radian measure.

Question 7. Find the angle in radians through which a pendulum swings, if its length is 75 cm and the tip describes an arc of length:
(i) 10 cm
(ii) 15 cm
(iii) 21 cm
Answer: The length of the pendulum acts as the radius \( r \), and the arc described by the tip is the arc length \( l \). We use the formula \( \theta = \frac{l}{r} \) to find the angle \( \theta \) in radians.
(i) For arc length \( l = 10 \) cm:
Given \( r = 75 \) cm, \( l = 10 \) cm.
\( \implies \theta = \frac{l}{r} = \frac{10}{75} \) radians. This fraction simplifies by dividing both numerator and denominator by 5.
\( \implies \theta = \frac{2}{15} \) radians.
(ii) For arc length \( l = 15 \) cm:
Given \( r = 75 \) cm, \( l = 15 \) cm.
\( \implies \theta = \frac{l}{r} = \frac{15}{75} \) radians. This fraction simplifies by dividing both numerator and denominator by 15.
\( \implies \theta = \frac{1}{5} \) radians.
(iii) For arc length \( l = 21 \) cm:
Given \( r = 75 \) cm, \( l = 21 \) cm.
\( \implies \theta = \frac{l}{r} = \frac{21}{75} \) radians. This fraction simplifies by dividing both numerator and denominator by 3.
\( \implies \theta = \frac{7}{25} \) radians.
In simple words: The angle a pendulum swings is found by dividing the arc length (how far the tip moves) by the length of the pendulum. The answer will be in radians.

Exam Tip: For pendulum questions, the length of the pendulum is the radius, and the distance its tip moves is the arc length. Always provide the answer in radians unless specified otherwise.