Exercise 3.1 — Trigonometric Functions

1. 3.1.13 marksNCERT Class 11 Mathematics, Trigonometric Functions, Reprint 2026-27

   Find the radian measures corresponding to the following degree measures: (i) 25° (ii) -47°30' (iii) 240° (iv) 520°.

   Hint. Multiply each degree measure by pi/180; convert minutes to a decimal fraction of a degree first.

   (i) 25 x pi/180 = 5pi/36. (ii) -47deg30' = -47.5deg = -95/2 deg; times pi/180 gives -19pi/72. (iii) 240 x pi/180 = 4pi/3. (iv) 520 x pi/180 = 26pi/9.

   ✦ Working through each part gives: (i) 5pi/36 (ii) -19pi/72 (iii) 4pi/3 (iv) 26pi/9.

2. 3.1.23 marksNCERT Class 11 Mathematics, Trigonometric Functions, Reprint 2026-27

   Find the degree measures corresponding to the following radian measures (use pi = 22/7): (i) 11/16 (ii) -4 (iii) 5pi/3 (iv) 7pi/6.

   Hint. Multiply each radian measure by 180/pi, using 180/pi = 630/11 when pi=22/7.

   180/pi = 1260/22 = 630/11 with pi=22/7. (i) 11/16 x 630/11 = 630/16 = 39.375deg = 39deg22'30". (ii) -4 x 630/11 = -2520/11 = -229 1/11 deg = -229deg5'27" approx. (iii) 5pi/3 x 180/pi = 300deg. (iv) 7pi/6 x 180/pi = 210deg.

   ✦ Working through each part gives: (i) 39deg22'30" (ii) -229deg5'27" approx (iii) 300deg (iv) 210deg.

3. 3.1.32 marksNCERT Class 11 Mathematics, Trigonometric Functions, Reprint 2026-27

   A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

   Hint. Convert revolutions per minute to revolutions per second first, then multiply by 2pi radians per revolution.

   360 revolutions per minute = 360/60 = 6 revolutions per second. Each revolution is 2pi radians, so in one second the wheel turns through 6 x 2pi = 12pi radians.

   ✦ Working through each part gives: 12pi radians.

4. 3.1.42 marksNCERT Class 11 Mathematics, Trigonometric Functions, Reprint 2026-27

   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 = 22/7).

   Hint. Find theta in radians first using theta = l/r, then convert to degrees.

   theta = l/r = 22/100 = 0.22 radian. Degree measure = 0.22 x 180/pi = 0.22 x 630/11 = 138.6/11 = 12.6deg = 12deg36'.

   ✦ Working through each part gives: 12deg36'.

5. 3.1.53 marksNCERT Class 11 Mathematics, Trigonometric Functions, Reprint 2026-27

   In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.

   Hint. The radius, the chord, and the two radii to its endpoints form a triangle -- check what kind of triangle it is before finding the central angle.

   Radius = 20 cm (half the diameter). Since the chord also equals 20 cm, the triangle formed by the chord and the two radii has all three sides equal to 20 cm -- an equilateral triangle, so the central angle is 60deg = pi/3 radians. Arc length = r x theta = 20 x pi/3 = 20pi/3 cm.

   ✦ Working through each part gives: 20pi/3 cm, approximately 20.94 cm.

   Where students slip. Assuming the central angle is 90 degrees (as if the chord were a side of a square) instead of checking that chord = radius forces an equilateral triangle and a 60 degree angle.

6. 3.1.62 marksNCERT Class 11 Mathematics, Trigonometric Functions, Reprint 2026-27

   If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.

   Hint. The arc length l = r-theta is the same for both circles -- set r1-theta1 = r2-theta2 and solve for the ratio.

   l = r1 theta1 = r2 theta2 (theta in radians, but the ratio works the same in degrees since a common factor of pi/180 cancels). So r1/r2 = theta2/theta1 = 75/60 = 5/4.

   ✦ Working through each part gives: r1 : r2 = 5 : 4.

7. 3.1.73 marksNCERT Class 11 Mathematics, Trigonometric Functions, Reprint 2026-27

   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) 15 cm (iii) 21 cm.

   Hint. Apply theta = l/r directly for each arc length, with r = 75 cm fixed throughout.

   theta = l/r in every part, with r=75 cm. (i) 10/75 = 2/15 radian. (ii) 15/75 = 1/5 radian. (iii) 21/75 = 7/25 radian.

   ✦ Working through each part gives: (i) 2/15 radian (ii) 1/5 radian (iii) 7/25 radian.

Solutions written by the tuition.in editorial team and checked against the NCERT Class 11 Mathematics textbook, Reprint 2026-27 (kemh103.pdf) — three numbered exercises (3.1-3.3, 42 questions) plus the chapter's Miscellaneous Exercise (10 questions); the chapter has no Exercise 3.4 and no general-solution-of-trigonometric-equations content, contrary to an earlier stub. Miscellaneous Exercise Q1's stacked-fraction identity was re-rendered directly from the PDF at 300dpi after raw text extraction garbled the numerators, confirming it as 2cos(pi/13)cos(9pi/13)+cos(3pi/13)+cos(5pi/13)=0. Questions are referenced from the NCERT textbook for identification.