NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.1

NCERT Solutions for Class 11 Maths Chapter 3 Exercise 3.1 focus on the foundational concepts of trigonometric functions, including the definition of trigonometric ratios in terms of a right-angled triangle, positive and negative values of trigonometric functions, and standard trigonometric values. This exercise helps students grasp the basics of trigonometric ratios and their applications in various geometrical contexts.

The exercise introduces students to the basic trigonometric ratios such as sine (sin), cosine (cos), tangent (tan), and others, along with their reciprocal functions. Students will also learn to express trigonometric functions for different angles and apply their knowledge in solving problems.

NCERT Solutions for Class 11 Maths Chapter 3 Trigonometric Functions Exercise 3.1

Trigonometric Functions

Medium

Q.

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

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Trigonometric Functions

Medium

Q.

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

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Trigonometric Functions

Medium

Q.

Find the value of: (i) sin75°    (ii) tan15°

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Trigonometric Functions

Medium

Q.

$\cos (\frac{\mathrm{\pi }}{4}-\mathrm{x})\cos (\frac{\mathrm{\pi }}{4}-\mathrm{y})-\sin (\frac{\mathrm{\pi }}{4}-\mathrm{x})\sin (\frac{\mathrm{\pi }}{4}-\mathrm{y})=\sin (\mathrm{x}+\mathrm{y})$

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Trigonometric Functions

Difficult

Q.

$\begin{array}{l}\text{For any triangle ABC},\text{ prove that}\\ \frac{{\mathrm{b}}^2-{\mathrm{c}}^2}{{\mathrm{a}}^2}\sin 2\mathrm{A}+\frac{{\mathrm{c}}^2-{\mathrm{a}}^2}{{\mathrm{b}}^2}\sin 2\mathrm{B}+\frac{{\mathrm{a}}^2-{\mathrm{b}}^2}{{\mathrm{c}}^2}\sin 2\mathrm{C}=0\end{array}$

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Trigonometric Functions

Difficult

Q.

A tree stands vertically on a hill side which makes an angle of 15° with the horizontal. From a point on the ground 35 m down the hill from the base of the tree, the angle of elevation of the top of the tree is 60°. Find the height of the tree.

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Trigonometric Functions

Difficult

Q.

Two ships leave a port at the same time. One goes 24 km per hour in the direction N45°E and other travels 32 km per hour in the direction S75°E. Find the distance between the ships at the end of 3 hours.

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Trigonometric Functions

Medium

Q.

$\begin{array}{l}\mathrm{Two} \mathrm{trees}, \mathrm{A} \mathrm{and} \mathrm{B} \mathrm{are} \mathrm{on} \mathrm{the} \mathrm{same} \mathrm{side} \mathrm{of} \mathrm{a} \mathrm{river}. \mathrm{From} \mathrm{a}\\ \mathrm{point} \mathrm{C} \mathrm{in} \mathrm{the} \mathrm{river} \mathrm{the} \mathrm{distance} \mathrm{of} \mathrm{the} \mathrm{trees} \mathrm{A} \mathrm{and} \mathrm{B} \mathrm{is} 250 \mathrm{m}\\ \mathrm{and} 300 \mathrm{m}, \mathrm{respectively}. \mathrm{If} \mathrm{the} \mathrm{angle} \mathrm{C} \mathrm{is} 45\mathrm{°}, \mathrm{find} \mathrm{the}\\ \mathrm{distance} \mathrm{between} \mathrm{the} \mathrm{trees} (\mathrm{use} \sqrt{2}=1.44).\end{array}$

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Trigonometric Functions

Medium

Q.

Prove that:
${(\cos \mathrm{x}-\cos \mathrm{y})}^2+{(\sin \mathrm{x}-\sin \mathrm{y})}^2=4 {\sin }^2 \frac{\mathrm{x}-\mathrm{y}}{2}$

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Question 1: Find the radian measures for the following degree measures:

• (i) 25°
  • Ans: Hum jante hain ki 180 degree = pi radian hota hai. Isliye, 25 degree = (pi / 180) x 25 = 5pi / 36 radian.
• (ii) -47° 30'
  • Ans: Pehle ise degree mein badlenge: -47 degree 30 minutes = -47.5 degree ya -95/2 degree. Ab ise radian mein badalne par: (pi / 180) x (-95/2) = -19pi / 72 radian.
• (iii) 240°
  • Ans: 240 degree = (pi / 180) x 240 = 4pi / 3 radian.
• (iv) 520°
  • Ans: 520 degree = (pi / 180) x 520 = 26pi / 9 radian.

Question 2: Find the degree measures for the following radian measures (Use pi = 22/7):

• (i) 11/16
  • Ans: (180 / pi) x (11/16) = 315 / 8 degree. Ise simplify karne par: 39 degree 22' 30".
• (ii) -4
  • Ans: (180 / pi) x (-4) = -2520 / 11 degree. Ise simplify karne par: -229 degree 5' 27".
• (iii) 5pi / 3
  • Ans: (180 / pi) x (5pi / 3) = 300 degree.
• (iv) 7pi / 6
  • Ans: (180 / pi) x (7pi / 6) = 210 degree.

Question 3: A wheel makes 360 revolutions in one minute. Through how many radians does it turn in one second?

• Ans: Ek minute (60 seconds) mein revolutions = 360. Ek second mein revolutions = 360 / 60 = 6. Ek revolution mein wheel 2pi radian ghumta hai. Toh 6 revolutions mein wheel 12pi radian ghumega.

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 = 22/7)

• Ans: Formula: theta = arc length (l) / radius (r). Yahan l = 22 cm aur r = 100 cm hai. Theta = 22 / 100 radian. Degree mein badalne par: (180 / pi) x (22 / 100) = 12 degree 36'.

Question 5: In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.

• Ans: Diameter = 40 cm, toh Radius (r) = 20 cm. Chord ki length bhi 20 cm hai, isliye triangle OAB ek equilateral triangle hai. Iska matlab angle (theta) = 60 degree ya pi/3 radian hai. Arc length (l) = r x theta = 20 x (pi/3) = 20pi / 3 cm.

Question 6: If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.

• Ans: Let radii be r1 and r2. Angles are 60 degree (pi/3) aur 75 degree (5pi/12). Kyunki arc length (l) barabar hai: r1 x (pi/3) = r2 x (5pi/12). Isse r1/r2 = 5/4 milta hai. Ratio 5:4 hai.

Question 7: 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: theta = 10 / 75 = 2/15 radian.
• (ii) 15 cm: theta = 15 / 75 = 1/5 radian.
• (iii) 21 cm: theta = 21 / 75 = 7/25 radian

FAQs – Class 11 Maths Chapter 3 Exercise 3.1 Trigonometric Functions

Q1. What is the focus of Exercise 3.1?
Exercise 3.1 focuses on the definition of trigonometric ratios and trigonometric functions of angles. It also involves the evaluation of standard trigonometric values for angles like 0°, 30°, 45°, 60°, and 90°.

Q2. What are the basic trigonometric ratios?
The basic trigonometric ratios are:

• Sine (sin) = Opposite / Hypotenuse

• Cosine (cos) = Adjacent / Hypotenuse

• Tangent (tan) = Opposite / Adjacent

• Cosecant (csc) = 1 / sin

• Secant (sec) = 1 / cos

• Cotangent (cot) = 1 / tan

Q3. Why is Exercise 3.1 important for board exams?
Exercise 3.1 is important because it provides a solid foundation in trigonometric ratios and their standard values, which are essential for solving problems in higher trigonometry, calculus, and other related topics. These concepts are frequently tested in Class 11 exams.

Q4. How can students prepare effectively for Exercise 3.1?
Students should:

• Memorize the trigonometric ratios and their reciprocal functions.

• Learn the standard trigonometric values for common angles (0°, 30°, 45°, 60°, 90°).

• Practice solving problems involving the evaluation of trigonometric functions for different angles.

Q5. How are trigonometric functions used in real-life applications?
Trigonometric functions are used in various fields such as engineering, physics, architecture, and navigation, especially when dealing with angles, heights, distances, and periodic phenomena.