NCERT Class 11 Maths Exercise 3.1 Solutions – Trigonometric Functions (Chapter 3)

Concept Theory

In trigonometry, angles can be measured in two standard units: degrees and radians.

A complete revolution of a circle corresponds to:

360° in degree measure
2π radians in radian measure

Hence the fundamental conversion relation becomes

\(180^\circ = \pi \text{ radians}\)

Therefore,

\(1^\circ = \dfrac{\pi}{180} \text{ radians}\)

To convert any angle from degrees to radians, we multiply the degree measure by \(\dfrac{\pi}{180}\).

The radian measure relates angles directly to the arc length of the circle.

Solution Roadmap

To solve this problem efficiently:

Use the relation \(1^\circ = \dfrac{\pi}{180}\).
Convert degrees (and minutes if present) into decimal degrees.
Multiply by \(\dfrac{\pi}{180}\).
Simplify the fraction.

Solution

We use the relation

\(1^\circ = \dfrac{\pi}{180}\) radians

(i) 25°

\[ \begin{aligned} 25^\circ &= 25 \times \frac{\pi}{180} \\ &= \frac{25\pi}{180} \\ &= \frac{5\pi}{36} \end{aligned} \]

(ii) –47°30′

First convert minutes into degrees:

\[ \begin{aligned} 30^\prime &= \frac{30}{60}^\circ \\ &= 0.5^\circ \end{aligned} \] \[ \begin{aligned} -47^\circ 30^\prime = -47.5^\circ \end{aligned} \] Now convert to radians: \[ \begin{aligned} -47.5^\circ &= -47.5 \times \frac{\pi}{180} \\ &= -\frac{47.5\pi}{180} \\ &= -\frac{95\pi}{360} \\ &= -\frac{19\pi}{72} \end{aligned} \]

(iii) 240°

\[ \begin{aligned} 240^\circ &= 240 \times \frac{\pi}{180} \\ &= \frac{240\pi}{180} \\ &= \frac{4\pi}{3} \end{aligned} \]

(iv) 520°

\[ \begin{aligned} 520^\circ &= 520 \times \frac{\pi}{180} \\ &= \frac{520\pi}{180} \\ &= \frac{26\pi}{9} \end{aligned} \]

Hence the radian measures are

\( \frac{5\pi}{36}, \; -\frac{19\pi}{72}, \; \frac{4\pi}{3}, \; \frac{26\pi}{9} \)

Exam Significance

Understanding degree–radian conversion is fundamental in trigonometry and appears frequently in board examinations and competitive entrance tests.

CBSE Board Exams: Direct conversion questions and simplification problems are common.
JEE Main & Advanced: Radian measure is essential for solving trigonometric identities, limits, derivatives, and unit circle problems.
NEET / BITSAT / CUET: Many trigonometric formulae are naturally expressed in radians, making this conversion indispensable.

Mastering this concept ensures a strong foundation for the entire study of trigonometric functions.

Concept Theory

Angles can be measured either in degrees or radians. A complete circle represents

360° in degree measure
2π radians in radian measure

From this relationship we obtain the fundamental conversion formula

\[ \pi \text{ radians} = 180^\circ \]

Therefore,

\[ 1 \text{ radian} = \frac{180^\circ}{\pi} \]

To convert radians into degrees, we multiply the radian measure by \(\dfrac{180}{\pi}\).

Radians measure the angle subtended by an arc equal to the radius of the circle.

Solution Roadmap

To convert radians into degrees:

Use the conversion \(1\text{ rad} = \dfrac{180}{\pi}^\circ\).
Substitute \(\pi = \dfrac{22}{7}\) as instructed.
Multiply the radian measure by \(\dfrac{180}{\pi}\).
Simplify and convert decimal degrees into minutes and seconds when required.

Solution

Using

\[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]

(i) \( \dfrac{11}{16} \)

\[ \begin{aligned} \dfrac{11}{16} &= \dfrac{180}{\pi} \times \dfrac{11}{16} \\ &= \dfrac{180 \times 7}{22} \times \dfrac{11}{16} \\ &= \dfrac{90 \times 7}{16} \\ &= \dfrac{630}{16} \\ &= 39.375^\circ \end{aligned} \] Convert to minutes: \[ 0.375 \times 60 = 22.5^\prime \] \[ 0.5 \times 60 = 30^{\prime\prime} \] \[ 39^\circ 22^{\prime} 30^{\prime\prime} \]

(ii) \(-4\)

\[ \begin{aligned} -4 &= \dfrac{180}{\pi} \times (-4) \\ &= \dfrac{180 \times 7}{22} \times (-4) \\ &= -\dfrac{2520}{11} \\ &= -229.0909^\circ \end{aligned} \] Convert fractional part: \[ 0.0909 \times 60 = 5.454^\prime \] \[ 0.454 \times 60 \approx 2^{\prime\prime} \] \[ -229^\circ 5^\prime 27^{\prime\prime} \]

(iii) \( \dfrac{5\pi}{3} \)

\[ \begin{aligned} \dfrac{5\pi}{3} &= \dfrac{180}{\pi} \times \dfrac{5\pi}{3} \\ &= \dfrac{180 \times 5}{3} \\ &= 300^\circ \end{aligned} \]

(iv) \( \dfrac{7\pi}{6} \)

\[ \begin{aligned} \dfrac{7\pi}{6} &= \dfrac{180}{\pi} \times \dfrac{7\pi}{6} \\ &= 30 \times 7 \\ &= 210^\circ \end{aligned} \]

Required degree measures:

\(39^\circ 22^\prime 30^{\prime\prime}\), \(-229^\circ 5^\prime 27^{\prime\prime}\), \(300^\circ\), \(210^\circ\)

Exam Significance

CBSE Board Exams: Questions often test conversion between radians and degrees along with angle simplification.
JEE Main / Advanced: Radian measure is the standard unit in calculus and trigonometric limits, making this conversion concept fundamental.
NEET / BITSAT / CUET: Many trigonometric identities and periodicity problems require quick conversion between radians and degrees.

A strong grasp of radian–degree conversion forms the foundation for studying trigonometric functions, graphs, and calculus.

Concept Theory

Rotational motion is commonly measured in revolutions or radians. A key relation between them is:

\[ 1 \text{ revolution} = 2\pi \text{ radians} \]

This follows from the fact that a full circle measures \(360^\circ\), which is equivalent to \(2\pi\) radians.

Therefore, when a rotating object completes several revolutions, the total angle turned can be found by multiplying the number of revolutions by \(2\pi\).

Each complete rotation of the wheel corresponds to an angular displacement of \(2\pi\) radians.

Solution Roadmap

To determine the radians turned in one second:

Convert revolutions per minute into revolutions per second.
Use the relation \(1 \text{ revolution} = 2\pi \text{ radians}\).
Multiply the revolutions per second by \(2\pi\).

Solution

The wheel makes

\[ 360 \text{ revolutions per minute} \]

Since

\[ 1 \text{ minute} = 60 \text{ seconds} \]

Revolutions made in one second:

\[ \begin{aligned} \text{Revolutions per second} &= \frac{360}{60} \\ &= 6 \end{aligned} \]

We know that

\[ 1 \text{ revolution} = 2\pi \text{ radians} \]

Therefore, the angular displacement in one second is

\[ \begin{aligned} \text{Radians turned in one second} &= 6 \times 2\pi \\ &= 12\pi \end{aligned} \]

Hence, the wheel turns through \(12\pi\) radians in one second.

Exam Significance

CBSE Board Exams: Questions involving revolutions, angular displacement, and radian conversion are common in trigonometry chapters.
JEE Main / Advanced: Rotational motion and angular displacement frequently appear in trigonometry, circular motion, and calculus problems.
NEET / BITSAT: Understanding radians is essential for solving periodic function and angular velocity problems.

Mastering the relation between revolutions and radians helps students connect trigonometry with real-world rotational motion.

Concept Theory

In a circle, the relationship between arc length, radius, and central angle is fundamental in radian measure.

If

\(s\) = length of the arc
\(r\) = radius of the circle
\(\theta\) = central angle in radians

then the arc length formula is

\( s = r\theta \)

Thus the central angle can be obtained as

\[ \theta = \frac{s}{r} \]

Once the angle is obtained in radians, it can be converted into degrees using

\[ \theta^\circ = \theta \times \frac{180}{\pi} \]

The central angle depends on the ratio of arc length to radius.

Solution Roadmap

To determine the required angle:

Use the arc-length formula \( \theta = \dfrac{s}{r} \) to find the angle in radians.
Substitute the given values \(s = 22\) cm and \(r = 100\) cm.
Convert the result from radians to degrees using \( \dfrac{180}{\pi} \).
Use \( \pi = \dfrac{22}{7} \) as instructed.

Solution

Given:

\[ r = 100\ \text{cm}, \qquad s = 22\ \text{cm} \]

Using

\[ \theta = \frac{s}{r} \] \[ \theta = \frac{22}{100} = 0.22\ \text{radians} \]

Convert radians into degrees:

\[ \begin{aligned} \theta^\circ &= 0.22 \times \frac{180}{\pi} \\ &= \frac{22}{100} \times \frac{180}{22/7} \\ &= \frac{22}{100} \times \frac{180 \times 7}{22} \\ &= \frac{1260}{100} \\ &= 12.6^\circ \end{aligned} \]

Convert the decimal part into minutes:

\[ 0.6 \times 60 = 36^\prime \]

Therefore, the required angle is

\(12^\circ 36^\prime\)

Exam Significance

CBSE Board Exams: Arc length and central angle problems are frequently asked to test understanding of radian measure.
JEE Main / Advanced: The relation \(s = r\theta\) is essential in trigonometry, circular motion, and calculus problems.
NEET / BITSAT: Angular displacement and arc-length relations often appear in physics and trigonometry-based questions.

Mastering the arc length formula builds a strong foundation for advanced topics such as circular motion, trigonometric functions, and calculus.

Concept Theory

In a circle, a chord subtends an angle at the centre. The relationship between the chord length \(c\), radius \(r\), and the central angle \(\theta\) is

\(c = 2r\sin(\theta/2)\)

Once the central angle is known, the corresponding arc length can be found using

\(s = r\theta\)

where \(\theta\) must be expressed in radians.

The chord subtends an angle at the centre which determines the arc length.

Solution Roadmap

To determine the length of the minor arc:

Determine the radius of the circle.
Use the chord–angle relation to find the central angle.
Convert the angle to radians.
Use the arc length formula \(s=r\theta\).

Solution

Diameter of the circle = 40 cm

\[ r = 20\text{ cm} \]

Length of chord

\[ c = 20\text{ cm} \]

Let the chord subtend angle \(2\theta\) at the centre.

Drop a perpendicular from the centre to the chord. Half of the chord becomes

\[ \frac{20}{2} = 10\text{ cm} \] Thus, \[ \sin\theta = \frac{10}{20} = \frac{1}{2} \] \[ \theta = 30^\circ \] Hence the central angle \[ 2\theta = 60^\circ \]

Convert into radians:

\[ 60^\circ = \frac{\pi}{3} \]

Arc length:

\[ \begin{aligned} s &= r\theta \\ &= 20 \times \frac{\pi}{3} \\ &= \frac{20\pi}{3}\text{ cm} \end{aligned} \]

Length of the minor arc = \( \frac{20\pi}{3} \) cm

Exam Significance

CBSE Board Exams: Problems involving chord length, central angle, and arc length are frequently asked.
JEE Main / Advanced: Understanding the relation between chord, radius, and angle is important in trigonometry and coordinate geometry.
NEET / BITSAT: Arc length and angular displacement concepts also appear in physics problems related to circular motion.

Mastering the connection between chords, angles, and arcs strengthens conceptual understanding of circle geometry and trigonometry.

Concept Theory

The length of an arc of a circle depends on the radius of the circle and the angle subtended by the arc at the centre.

If

\(s\) = arc length
\(r\) = radius of the circle
\(\theta\) = central angle in radians

then the arc length relation is

\( s = r\theta \)

For arcs of the same length, the radii of the circles are inversely proportional to the angles subtended at their centres.

Equal arc lengths in different circles correspond to different central angles depending on the radius.

Solution Roadmap

To determine the ratio of the radii:

Use the arc length relation \(s = r\theta\).
Since arc lengths are equal, equate the two expressions.
Solve the equation to obtain the ratio of radii.

Solution

Let

\(r_1\) = radius of the first circle
\(r_2\) = radius of the second circle

Angles subtended at the centres:

\[ \theta_1 = 60^\circ, \qquad \theta_2 = 75^\circ \]

Arc length formula:

\[ s = r\theta \] For the first circle \[ s_1 = r_1 \theta_1 \] For the second circle \[ s_2 = r_2 \theta_2 \]

Since the arcs are of equal length:

\[ s_1 = s_2 \] \[ r_1 \theta_1 = r_2 \theta_2 \]

Therefore

\[ \frac{r_1}{r_2} = \frac{\theta_2}{\theta_1} \] \[ = \frac{75}{60} \] \[ = \frac{5}{4} \]

Hence, the ratio of the radii is

\(r_1 : r_2 = 5 : 4\)

Exam Significance

CBSE Board Exams: Problems testing arc length relations and proportional reasoning frequently appear.
JEE Main / Advanced: Understanding how radius and angle affect arc length is important for trigonometry and circular geometry problems.
NEET / BITSAT: Similar concepts are used in circular motion and angular displacement problems.

Recognizing that arc length depends on both radius and angle helps solve many geometry and trigonometry problems efficiently.

Concept Theory

When a pendulum swings, its tip moves along a circular arc. The length of the pendulum acts as the radius of the circular path.

If

\(s\) = arc length
\(r\) = radius (length of pendulum)
\(\theta\) = angle in radians

then the relation between them is

\( s = r\theta \)

Hence,

\[ \theta = \frac{s}{r} \]

Thus the angle through which the pendulum swings depends on the ratio of arc length to the length of the pendulum.

The pendulum tip moves along a circular arc whose radius equals the length of the pendulum.

Solution Roadmap

To find the angle of swing:

Use the relation \( \theta = \dfrac{s}{r} \).
Substitute the arc length \(s\) and radius \(r = 75\) cm.
Simplify the fraction to obtain the angle in radians.

Solution

Length of pendulum (radius)

\[ r = 75\ cm \]

(i) Arc length = 10 cm

\[ \begin{aligned} \theta &= \frac{s}{r} \\ &= \frac{10}{75} \\ &= \frac{2}{15} \end{aligned} \] Angle of swing: \[ \frac{2}{15} \text{ radians} \]

(ii) Arc length = 15 cm

\[ \begin{aligned} \theta &= \frac{15}{75} \\ &= \frac{1}{5} \end{aligned} \] Angle of swing: \[ \frac{1}{5} \text{ radians} \]

(iii) Arc length = 21 cm

\[ \begin{aligned} \theta &= \frac{21}{75} \\ &= \frac{7}{25} \end{aligned} \] Angle of swing: \[ \frac{7}{25} \text{ radians} \]

Hence, the required angles are

\( \frac{2}{15},\; \frac{1}{5},\; \frac{7}{25} \) radians

Exam Significance

CBSE Board Exams: Pendulum and arc length problems frequently appear to test understanding of radian measure.
JEE Main / Advanced: The relation \(s = r\theta\) is widely used in trigonometry, circular motion, and calculus.
NEET / BITSAT: Similar concepts appear in physics questions involving angular displacement and oscillatory motion.

Understanding the connection between arc length and angle helps students interpret many real-world circular motion problems.

Chapter 3 — TRIGONOMETRIC FUNCTIONS

Concept Theory

Solution Roadmap

Solution

Exam Significance

Concept Theory

Solution Roadmap

Solution

Exam Significance

Concept Theory

Solution Roadmap

Solution

Exam Significance

Concept Theory

Solution Roadmap

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Exam Significance

Concept Theory

Solution Roadmap

Solution

Exam Significance

Concept Theory

Solution Roadmap

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Exam Significance

Concept Theory

Solution Roadmap

Solution

Exam Significance

Chapter Complete!

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