We know that Trigonometry is a branch of Mathematics which deals with the measurement of the sides and the angles of a triangle and the problems allied with angles. In this chapter, we mostly learn about the relation between degrees, radians and real numbers. Expert tutors at BYJUâ€™S have designed the RD Sharma Class 11 Solutions in order to help students to gain more knowledge and create a strong grip over the subject. Students are advised to practice the solutions regularly so that they can come out with flying colours in their exams. Solutions to this chapter are provided in the pdf format, which can be downloaded easily from the links provided below.

Chapter 4- Measurement of Angles contains one exercise and the RD Sharma Solutions present in this page provide solutions to the questions present in this exercise. Now, let us have a look at the concepts discussed in this chapter.

- Angles.
- Some useful terms.
- Systems of measurement of angles.
- Sexagesimal system.
- Centesimal system.
- A circular system.

- The relation between degrees and radians.
- The relation between radians and real numbers.
- The relation between three systems of measurement of an angle.

## Download the Pdf of RD Sharma Solutions for Class 11 Maths Chapter 4 – Measurement of Angles

### Access answers to RD Sharma Solutions for Class 11 Maths Chapter 4 – Measurement of Angles

**1. Find the degree measure corresponding to the following radian measures (Use Ï€ = 22/7) (i) 9Ï€/5Â (ii) -5Ï€/6Â (iii) (18Ï€/5) ^{ c}Â (iv) (-3)^{ c}Â (v) 11^{c}Â (vi) 1^{c}**

**Solution:**

We know that Ï€ rad = 180Â°Â â‡’Â 1 rad = 180Â°/ Ï€

**(i)** Â 9Ï€/5Â

^{ o }

Substituting the value of Ï€ = 22/7 [180/22 Ã— 7 Ã— 9 Ã— 22/(7Ã—5)]

(36 Ã— 9) Â°

324Â°

âˆ´ Degree measure of 9Ï€/5Â is 324Â°

**(ii)** -5Ï€/6**Â **

^{ o }

Substituting the value of Ï€ = 22/7 [180/22 Ã— 7 Ã— -5 Ã— 22/(7Ã—6) ]

(30 Ã— -5) Â°

– (150) Â°

âˆ´ Degree measure of -5Ï€/6Â is -150Â°

**(iii)** (18Ï€/5)

^{ o }

Substituting the value of Ï€ = 22/7 [180/22 Ã— 7 Ã— 18 Ã— 22/(7Ã—5)]

(36 Ã— 18) Â°

648Â°

âˆ´ Degree measure of 18Ï€/5Â is 648Â°

**(iv) **(-3)^{ c}**Â **

^{ o}

Substituting the value of Ï€ = 22/7

[180/22 Ã— 7 Ã— -3]^{ o}

(-3780/22)^{ o}

(-171 18/22)^{ o}

(-171^{ o }(18/22 Ã— 60)’)

(-171^{o} (49 1/11)’)

(-171^{o} 49′ (1/11 Ã— 60)’)

– (171Â° 49′ 5.45”)

â‰ˆ – (171Â° 49′ 5”)

âˆ´ Degree measure of (-3)^{ c}**Â **is -171Â° 49′ 5”

**(v)** 11^{c}

(180/ Ï€ Ã— 11)^{ o}

Substituting the value of Ï€ = 22/7

(180/22 Ã— 7 Ã— 11)^{ o}

(90 Ã— 7) Â°

630Â°

âˆ´ Degree measure of 11^{c} is 630Â°

**(vi) **1^{c}

(180/ Ï€ Ã— 1)^{ o}

Substituting the value of Ï€ = 22/7

(180/22 Ã— 7 Ã— 1)^{ o}

(1260/22) ^{o}

(57 3/11) ^{o}

(57^{o} (3/11 Ã— 60)’)

(57^{o} (16 4/11)’)

(57^{o} 16′ (4/11 Ã— 60)’)

(57^{o} 16′ 21.81”)

â‰ˆ (57^{o} 16′ 21”)

âˆ´ Degree measure of 1^{c} is 57^{o} 16′ 21”

**2. Find the radian measure corresponding to the following degree measures:(i) 300 ^{o}Â (ii) 35^{o}Â (iii) -56^{o}Â (iv)135^{o}Â (v) -300^{o}(vi) 7^{o}Â 30′ (vii) 125^{o}Â 30â€™ (viii) -47^{o}Â 30′**

**Solution:**

We know that 180Â° = Ï€ radÂ â‡’Â 1Â° = Ï€/ 180 rad

**(i)** 300Â°

(300 Ã— Ï€/180) rad

5Ï€/3

âˆ´ Radian measure of 300^{o} is 5Ï€/3

**(ii)** 35Â°

(35 Ã— Ï€/180) rad

7Ï€/36

âˆ´ Radian measure of 35^{o} is 7Ï€/36

**(iii)** -56Â°

(-56 Ã— Ï€/180) rad

-14Ï€/45

âˆ´ Radian measure of -56Â° is -14Ï€/45

**(iv)** 135Â°

(135 Ã— Ï€/180) rad

3Ï€/4

âˆ´ Radian measure of 135Â° is 3Ï€/4

**(v)** -300Â°

(-300 Ã— Ï€/180) rad

-5Ï€/3

âˆ´ Radian measure of -300Â° is -5Ï€/3

**(vi)** 7Â° 30′

We know that, 30′ = (1/2) Â°

7Â° 30′ = (7 1/2) Â°

= (15/2)^{ o}

= (15/2 Ã— Ï€/180) rad

= Ï€/24

âˆ´ Radian measure of 7Â° 30′ is Ï€/24

**(vii)** 125Â° 30′

We know that, 30′ = (1/2) Â°

125Â° 30â€™ = (125 1/2) Â°

= (251/2)^{ o}

= (251/2 Ã— Ï€/180) rad

= 251Ï€/360

âˆ´ Radian measure of 125Â° 30′ is 251Ï€/360

**(viii)** -47Â° 30′

We know that, 30′ = (1/2) Â°

-47Â° 30â€™ = – (47 1/2) Â°

= – (95/2)^{ o}

= – (95/2 Ã— Ï€/180) rad

= – 19Ï€/72

âˆ´ Radian measure of -47Â° 30′ is – 19Ï€/72

**3. The difference between the two acute angles of a right-angled triangle is 2Ï€/5 radians. Express the angles in degrees.**

**Solution:**

Given the difference between the two acute angles of a right-angled triangle is 2Ï€/5 radians.

We know that Ï€ rad = 180Â°Â â‡’Â 1 rad = 180Â°/ Ï€

Given:

2Ï€/5

(2Ï€/5 Ã— 180/ Ï€)^{ o}

Substituting the value of Ï€ = 22/7

(2Ã—22/(7Ã—5) Ã— 180/22 Ã— 7)

(2/5 Ã— 180) Â°

72Â°

Let one acute angle be xÂ° and the other acute angle be 90Â° – xÂ°.

Then,

xÂ° – (90Â° – xÂ°) = 72Â°

2xÂ° – 90Â° = 72Â°

2xÂ° = 72Â° + 90Â°

2xÂ° = 162Â°

xÂ° = 162Â°/ 2

xÂ° = 81Â° and

90Â° – xÂ° = 90Â° – 81Â°

= 9Â°

âˆ´ The angles are 81^{o} and 9^{o}

**4. One angle of a triangle is 2/3xÂ grades,Â and another is 3/2xÂ degrees while the third is Ï€x/75Â radians. Express all the angles in degrees.**

**Solution:**

Given:

One angle of a triangle is 2x/3 grades and another is 3x/2 degree while the third is Ï€x/75 radians.

We know that, 1 grad = (9/10)^{ o}Â

2/3x grad = (9/10) (2/3x)^{ o}

= 3/5x^{o}

We know that, Ï€ rad = 180Â°Â â‡’Â 1 rad = 180Â°/ Ï€

Given: Ï€x/75

(Ï€x/75 Ã— 180/Ï€)^{ o}

(12/5x)^{ o}

We know that, the sum of the angles of a triangle is 180Â°.

3/5x^{o} + 3/2x^{o} + 12/5x^{o} = 180^{o}

(6+15+24)/10x^{o} = 180^{o}

Upon cross-multiplication we get,

45x^{o} = 180^{o} Ã— 10^{o}

= 1800^{o}

x^{o} = 1800^{o}/45^{o}

= 40^{o}

âˆ´Â The angles of the triangle are:

3/5x^{o} = 3/5 Ã— 40^{o} = 24^{o}

3/2x^{o} = 3/2 Ã— 40^{o} = 60^{o}

12/5 x^{o} = 12/5 Ã— 40^{o} = 96^{o}

**5. Find the magnitude, in radians and degrees, of the interior angle of a regular:(i) Pentagon (ii) Octagon (iii) Heptagon (iv) Duodecagon.**

**Solution:**

We know that the sum of the interior angles of a polygon = (n â€“ 2) Ï€

And each angle of polygon = sum of interior angles of polygon / number of sidesÂ

Now, let us calculate the magnitude of

**(i)** Pentagon

Number of sides in pentagon = 5

Sum of interior angles of pentagon = (5 â€“ 2) Ï€ = 3Ï€

âˆ´Â Each angle of pentagon = 3Ï€/5 Ã— 180^{o}/ Ï€ = 108^{o}

**(ii)** Octagon

Number of sides in octagon = 8

Sum of interior angles of octagon = (8 â€“ 2) Ï€ = 6Ï€

âˆ´Â Each angle of octagon = 6Ï€/8 Ã— 180^{o}/ Ï€ = 135^{o}Â

**(iii)** Heptagon

Number of sides in heptagon = 7

Sum of interior angles of heptagon = (7 â€“ 2) Ï€ = 5Ï€

âˆ´Â Each angle of heptagon = 5Ï€/7 Ã— 180^{o}/ Ï€ = 900^{o}/7 = 128^{o} 34′ 17” Â

**(iv)** Duodecagon

Number of sides in duodecagon = 12

Sum of interior angles of duodecagon = (12 â€“ 2) Ï€ = 10Ï€

âˆ´Â Each angle of duodecagon = 10Ï€/12 Ã— 180^{o}/ Ï€ = 150^{o}

**6. The angles of a quadrilateral are in A.P., and the greatest angle is 120 ^{o}. Express the angles in radians.**

**Solution:**

Let the angles of quadrilateral be (a â€“ 3d) Â°, (a â€“ d) Â°, (a + d) Â° and (a + 3d) Â°.

We know that, the sum of angles of a quadrilateral is 360Â°.

a â€“ 3d + a â€“ d + a + d + a + 3d = 360Â°

4a = 360Â°

a = 360/4

= 90Â°

Given:

The greatest angle = 120Â°

a + 3d = 120Â°

90Â° + 3d = 120Â°

3d = 120Â° – 90Â°

3d = 30Â°

d = 30Â°/3

= 10^{o}

âˆ´ The angles are:

(a â€“ 3d) Â° = 90Â° – 30Â° = 60Â°

(a â€“ d) Â° = 90Â° – 10Â° = 80Â°

(a + d) Â° = 90Â° + 10Â° = 100Â°

(a + 3d) Â° = 120Â°

Angles of quadrilateral in radians:

(60 Ã— Ï€/180) rad = Ï€/3

(80 Ã— Ï€/180) rad = 4Ï€/9

(100 Ã— Ï€/180) rad = 5Ï€/9

(120 Ã— Ï€/180) rad = 2Ï€/3

**7. The angles of a triangle are in A.P., and the number of degrees in the least angle is to the number of degrees in the mean angle as 1:120. Find the angle in radians.**

**Solution:**

Let the angles of the triangle be (a â€“ d) Â°, aÂ° and (a + d) Â°.

We know that, the sum of the angles of a triangle is 180Â°.

a â€“ d + a + a + d = 180Â°

3a = 180Â°

a = 60Â°

Given:

Number of degrees in the least angle / Number of degrees in the mean angle = 1/120 Â

(a-d)/a = 1/120

(60-d)/60 = 1/120

(60-d)/1 = 1/2

120-2d = 1

2d = 119

d = 119/2

= 59.5

âˆ´ The angles are:

(a â€“ d) Â° = 60Â° â€“ 59.5Â° = 0.5Â°

aÂ° = 60Â°

(a + d) Â° = 60Â° + 59.5Â° = 119.5Â°

Â Angles of triangle in radians:

(0.5 Ã— Ï€/180) rad = Ï€/360

(60 Ã— Ï€/180) rad = Ï€/3

(119.5 Ã— Ï€/180) rad = 239Ï€/360

**8. The angle in one regular polygon is to that in another as 3:2 and the number of sides in first is twice that in the second. Determine the number of sides of two polygons.**

**Solution:**

Let the number of sides in the first polygon be 2x and

The number of sides in the second polygon be x.

We know that, angle of an n-sided regular polygon = [(n-2)/n] Ï€ radianÂ

The angle of the first polygonÂ = [(2x-2)/2x] Ï€ = [(x-1)/x] Ï€ radian

The angle of the second polygon = [(x-2)/x] Ï€ radian Â

Thus,

[(x-1)/x] Ï€ / [(x-2)/x] Ï€ = 3/2(x-1)/(x-2) = 3/2

Upon cross-multiplication we get,

2x â€“ 2 = 3x â€“ 6

3x-2x = 6-2

x = 4

âˆ´ Number of sides in the first polygon = 2x = 2(4) = 8

Number of sides in the second polygon = x = 4

**9. The angles of a triangle are in A.P. such that the greatest is 5 times the least. Find the angles in radians.**

**Solution:**

Let the angles of the triangle be (a â€“ d)^{ o}, a^{o} and (a + d)^{ o}.

We know that, the sum of angles of triangle is 180Â°.

a â€“ d + a + a + d = 180Â°

3a = 180Â°

a = 180Â°/3

= 60^{o}

Given:

Greatest angle = 5 Ã— least angle

Upon cross-multiplication,

Greatest angle / least angle = 5

(a+d)/(a-d) = 5

(60+d)/(60-d) = 5

By cross-multiplying we get,

60 + d = 300 â€“ 5d

6d = 240

d = 240/6

= 40

Hence, angles are:

(a â€“ d) Â° = 60Â° â€“ 40Â° = 20Â°

aÂ° = 60Â°

(a + d) Â° = 60Â° + 40Â° = 100Â°

âˆ´Â Angles of triangle in radians:

(20 Ã— Ï€/180) rad = Ï€/9

(60 Ã— Ï€/180) rad = Ï€/3

(100 Ã— Ï€/180) rad = 5Ï€/9

**10. The number of sides of two regular polygons is 5:4 and the difference between their angles is 9 ^{o}. Find the number of sides of the polygons.**

**Solution:**

Let the number of sides in the first polygon be 5x and

The number of sides in the second polygon be 4x.

We know that, angle of an n-sided regular polygon = [(n-2)/n] Ï€ radian

The angle of the first polygon = [(5x-2)/5x] 180^{o}

The angle of the second polygon = [(4x-1)/4x] 180^{o} Â

Thus,

[(5x-2)/5x] 180^{o}– [(4x-1)/4x] 180

^{o}= 9

180

^{o}[(4(5x-2) â€“ 5(4x-2))/20x] = 9

Upon cross-multiplication we get,

(20x â€“ 8 â€“ 20x + 10)/20x = 9/180

2/20x = 1/20

2/x = 1

x = 2

âˆ´Number of sides in the first polygon = 5x = 5(2) = 10

Number of sides in the second polygon = 4x = 4(2) = 8