Rbse Solutions for Class 11 maths Chapter 3 Exercise 3.1| Degree and Radian Measure

Get detailed, step-by-step solutions for NCERT Class 11 Maths Chapter 3 Exercise 3.1 . Master the conversion between Degree and Radian measure ($\pi/180$ and $180/\pi$). Practice using the formula $\mathbf{\theta = l/r}$ to find central angles, arc lengths, and radii in circles and pendulum problems. Includes complex conversions involving minutes (e.g., $-47^{\circ} 30’$). Essential for understanding the fundamentals of trigonometry.

This exercise focuses on the conversion between degree measure and radian measure and applying the relationship between arc length, radius, and central angle ($\theta = l/r$).

The conversion formulas are:

• Degrees to Radians: Multiply by $\frac{\pi}{180}$
• Radians to Degrees: Multiply by $\frac{180}{\pi}$

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1. Find the Radian Measures

(i) $25^{\circ}$

$$25^{\circ} \times \frac{\pi}{180} \text{ radians} = \frac{5\pi}{36} \text{ radians}$$

(ii) $-47^{\circ} 30’$

First, convert to decimal degrees: $30′ = (30/60)^{\circ} = 0.5^{\circ}$.

$$-47^{\circ} 30′ = -47.5^{\circ}$$

Convert to radians:

$$-47.5^{\circ} \times \frac{\pi}{180} \text{ radians} = -\frac{475\pi}{1800} \text{ radians}$$

$$-\frac{475\pi}{1800} = -\frac{19\pi}{72} \text{ radians}$$

(iii) $240^{\circ}$

$$240^{\circ} \times \frac{\pi}{180} \text{ radians} = \frac{24\pi}{18} \text{ radians} = \frac{4\pi}{3} \text{ radians}$$

(iv) $520^{\circ}$

$$520^{\circ} \times \frac{\pi}{180} \text{ radians} = \frac{52\pi}{18} \text{ radians} = \frac{26\pi}{9} \text{ radians}$$

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2. Find the Degree Measures (Use $\pi = 22/7$)

The conversion factor is $\frac{180}{\pi}$.

(i) $\frac{11}{16}$ radians

$$\frac{11}{16} \times \frac{180}{\pi} \text{ degrees} = \frac{11}{16} \times \frac{180}{22/7}$$

$$= \frac{11}{16} \times 180 \times \frac{7}{22} = \frac{1}{16} \times 180 \times \frac{7}{2}$$

$$= \frac{180 \times 7}{32} = \frac{45 \times 7}{8} = \frac{315}{8} \text{ degrees}$$

Convert to degrees, minutes, and seconds:

$$\frac{315}{8} = 39 \text{ degrees} \text{ with remainder } 3/8$$

$$39^{\circ} + \left(\frac{3}{8} \times 60\right)’ = 39^{\circ} + \frac{180}{8}’ = 39^{\circ} + 22.5’$$

$$39^{\circ} + 22′ + (0.5 \times 60)” = 39^{\circ} 22′ 30”$$

(ii) $-4$ radians

$$-4 \times \frac{180}{\pi} \text{ degrees} = -4 \times \frac{180}{22/7} = -4 \times 180 \times \frac{7}{22}$$

$$= -\frac{2 \times 180 \times 7}{11} = -\frac{2520}{11} \text{ degrees}$$

Convert to degrees, minutes, and seconds:

$$-\frac{2520}{11} = -(229 \text{ degrees} \text{ with remainder } 1/11)$$

$$-229^{\circ} – \left(\frac{1}{11} \times 60\right)’ = -229^{\circ} – \frac{60}{11}’ = -229^{\circ} – (5 \text{ minutes} \text{ with remainder } 5/11)$$

$$-229^{\circ} 5′ – \left(\frac{5}{11} \times 60\right)” \approx -229^{\circ} 5′ 27”$$

(iii) $\frac{5\pi}{3}$ radians

$$\frac{5\pi}{3} \times \frac{180}{\pi} \text{ degrees} = 5 \times 60^{\circ} = \mathbf{300^{\circ}}$$

(iv) $\frac{7\pi}{6}$ radians

$$\frac{7\pi}{6} \times \frac{180}{\pi} \text{ degrees} = 7 \times 30^{\circ} = \mathbf{210^{\circ}}$$

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3. Revolutions to Radians

A wheel makes 360 revolutions in 1 minute (60 seconds).

1. Revolutions per second:$$\text{Revolutions/second} = \frac{360 \text{ rev}}{60 \text{ sec}} = 6 \text{ revolutions/second}$$
2. Radians per revolution:One complete revolution equals $2\pi$ radians.
3. Radians per second:$$\text{Radians/second} = 6 \text{ rev/sec} \times 2\pi \text{ rad/rev} = \mathbf{12\pi \text{ radians}}$$

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4. Degree Measure of Central Angle

Given: Radius $r = 100 \text{ cm}$, Arc length $l = 22 \text{ cm}$. Use $\pi = 22/7$.

1. Find the angle in radians ($\theta$):$$\theta = \frac{l}{r} \text{ radians}$$$$\theta = \frac{22}{100} = \frac{11}{50} \text{ radians}$$
2. Convert $\theta$ to degrees:$$\theta = \frac{11}{50} \times \frac{180}{\pi} \text{ degrees} = \frac{11}{50} \times \frac{180}{22/7}$$$$\theta = \frac{11}{50} \times 180 \times \frac{7}{22} = \frac{1}{50} \times 180 \times \frac{7}{2}$$$$\theta = \frac{180 \times 7}{100} = \frac{18 \times 7}{10} = \frac{126}{10} = 12.6 \text{ degrees}$$
3. Convert to degrees and minutes:$$12.6^{\circ} = 12^{\circ} + (0.6 \times 60)’ = \mathbf{12^{\circ} 36′}$$

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5. Length of Minor Arc of a Chord

Given: Diameter $D = 40 \text{ cm}$, so Radius $r = 20 \text{ cm}$. Length of chord $c = 20 \text{ cm}$.

1. Identify the triangle: The triangle formed by the two radii and the chord (let’s call the centre $O$ and chord endpoints $A, B$) is $\triangle OAB$.
   • $OA = r = 20 \text{ cm}$
   • $OB = r = 20 \text{ cm}$
   • $AB = c = 20 \text{ cm}$
2. Find the central angle ($\theta$): Since all sides of $\triangle OAB$ are equal (20 cm), it is an equilateral triangle.The central angle $\angle AOB = \mathbf{60^{\circ}}$.
3. Convert $\theta$ to radians:$$\theta = 60^{\circ} \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}$$
4. Find the minor arc length ($l$):$$l = r \theta$$$$l = 20 \times \frac{\pi}{3} = \mathbf{\frac{20\pi}{3} \text{ cm}}$$

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6. Ratio of Radii for Equal Arc Lengths

Given: $l_1 = l_2 = l$. Angle 1: $\theta_1 = 60^{\circ}$. Angle 2: $\theta_2 = 75^{\circ}$. Radii: $r_1, r_2$.

1. Convert angles to radians:$$\theta_1 = 60^{\circ} \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}$$$$\theta_2 = 75^{\circ} \times \frac{\pi}{180} = \frac{5\pi}{12} \text{ radians}$$
2. Relate length and radius: $l = r\theta$, so $r = l/\theta$.$$r_1 = \frac{l}{\theta_1} = \frac{l}{\pi/3} = \frac{3l}{\pi}$$$$r_2 = \frac{l}{\theta_2} = \frac{l}{5\pi/12} = \frac{12l}{5\pi}$$
3. Find the ratio $r_1 : r_2$:$$\frac{r_1}{r_2} = \frac{3l/\pi}{12l/5\pi} = \frac{3l}{\pi} \times \frac{5\pi}{12l}$$$$\frac{r_1}{r_2} = \frac{3 \times 5}{12} = \frac{15}{12} = \frac{5}{4}$$The ratio of their radii is $\mathbf{5 : 4}$.

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7. Angle in Radian (Pendulum)

The path described by the pendulum tip is the arc length ($l$), and the pendulum’s length is the radius ($r$). The angle $\theta$ is found using $\theta = l/r$.

Given: Length $r = 75 \text{ cm}$.

(i) Arc length $l = 10 \text{ cm}$

$$\theta = \frac{10}{75} = \mathbf{\frac{2}{15} \text{ radians}}$$

(ii) Arc length $l = 15 \text{ cm}$

$$\theta = \frac{15}{75} = \mathbf{\frac{1}{5} \text{ radians}}$$

(iii) Arc length $l = 21 \text{ cm}$

$$\theta = \frac{21}{75} = \mathbf{\frac{7}{25} \text{ radians}}$$