Class 11 NCERT Solutions- Chapter 3 Trigonometric Function – Exercise 3.4

NCERT Solutions for Class 11 Maths Chapter 3 – Trigonometric Functions Exercise 3.4″ provides comprehensive answers and explanations to the questions and problems presented in Exercise 3.4 of the textbook.

This exercise focuses on exploring the properties and applications of trigonometric functions, including sine, cosine, and tangent, as well as their inverses. By following these solutions, students can deepen their understanding of trigonometry and enhance their problem-solving skills.

Solve Following Questions NCERT Solutions for Class 12 Math’s Chapter 1 (Trigonometric Function) – Exercise3.4

Find the Principal and General Solutions of Following Equations:

Question 1. tan x = √3

Solution:

Given: tan x = √3

Here, x lies in first or third quadrant.

∴ tanx = tan60°∘ or tanx = tan(180° + 60°)

tanx = tan60°∘ or tanx = tan240°∘

Here, the principal solutions are π/3​, 4π/3​.

∴ tanx = tanπ/3

⇒ x = nπ + π​/3 where n ∈ Z​

Question 2. sec x = 2

Solution:

Given: sec x = 2

It can be written as, cos x = 1/2

Here, x lies in first or fourth quadrant.

∴ cosx = cos60°∘ or cosx = cos(360° – 60°)

cosx = cos60°∘ or cosx = cos300°∘

cosx = cosπ/3 or cosx = cos 5π/3

Here, the principal solutions are π/3, 5 π/3

∴ cosx = cosπ

x = 2nπ ± π/3​ where, n ∈ Z​

Question 3. cot x = −√3

Solution:

Given: cot x = −√3

It can be written as, tan x = -1/√3

Here, x lies in second or fourth quadrant.

∴ tanx = −tan30°∘= tan(180°– 30°) or tanx = tan(360°∘- 60°)

tanx = tan150°∘ or tanx = tan330°∘

tanx = tan5π/6​ or tanx = tan11π/6​​

Here, principal solutions are 5π/6, 11 π/6

∴ tan x = tan 5π/6

⇒x = nπ + 5π/3​ where, n ∈ Z​

Question 4. cosec x = – 2

Solution:

Given: cosec x = – 2

It can be written as, sin x = -1/2

Here x lies in third or fourth quadrant.

∴ sinx = -sin30°∘= sin(180° + 30°) or sinx = sin(360°∘- 30°)

sinx = sin210°∘ or sinx = sin330°∘

sinx = sin7π/6​ or sinx = sin11π/6​​

Here, the principal solutions are 7π/6, 11 π/6

∴ sin x = -sin π/6

⇒ x = nπ + (−1)ⁿ7π/6​ where, n ∈ Z​

Find the General Solution for Each of the Following Equations:

Question 5. cos 4x = cos 2x

Solution:

Given: cos 4x = cos 2x

4x = 2nπ ± 2x, n ∈ Z

4x – 2x = 2nπ or 4x + 2x = 2nπ, n ∈ Z

2x = 2nπ or 6x = 2nπ, n ∈ Z

2x = 2nπ or 6x = 2nπ, n ∈ Z

x = nπ or x = 3nπ​, n ∈ Z

Therefore, the principal solutions are nπ, nπ​/3
 

Question 6. cos 3x + cos x – cos 2x = 0

Solution:

Given: cos 3x + cos x – cos 2x = 0

2cos2xcosx – cos2x = 0

cos2x(2cosx – 1) = 0

cos2x = 0 or 2cosx – 1 = 0

2x = (2n + 1)π/2​ or cosx = 1/2 ​= cosπ/3​, n ∈ z

x = (2n + 1)π/4​ or x = 2nπ ± 2π​/3, n ∈ z

Question 7. sin 2x + cos x = 0

Solution:

Given: sin 2x + cos x = 0

2sinxcosx + cosx = 0

cosx(2sinx + 1) = 0

cosx = 0 or 2sinx + 1 = 0

x = (2n + 1)π/2​ or sinx = −1/2​ = −sinπ/6​, n ∈ z

x = (2n + 1)π/4​ or x = 2nπ ± 2π/3​, n ∈ z

x = (2n + 1)π/4​ or x = nπ + (-1)ⁿ-π/6​, n ∈ Z

x = (2n + 1)π/4​ or x = nπ + (-1)ⁿ7π/6​, n ∈ Z

Question 8. sec²2x = 1 – tan 2x

Solution:

Given: sec²2x = 1 – tan 2x

1 + tan²2x = 1 – tan2x

tan2x(tan2x + 1) = 0

tan2x = 0 or tan2x + 1 = 0

2x = nπ or tan2x – 1 = –tanπ/4

x = nπ/2​ or x = nπ​/2 + 3π/8​, n = Z​
 

Question 9. sin x + sin 3x + sin 5x = 0

Solution:

Given: sin x + sin 3x + sin 5x = 0

2sin3xcos2x + sin3x = 0

sin3x(2cos2x + 1) = 0

sin3x = 0 or 2cos2x + 1 = 1

3x = nπ or cos2x = -1/2 ​= cos2π/3​, n ∈ z

x = nπ/3​ or 2x = nπ/2 ± 2π/3​, n ∈ z

x = nπ​/3 or x = nπ ± π/3​, n ∈ z

Trigonometric Functions Exercises:

What are trigonometric functions?

Trigonometric functions are mathematical functions that relate angles of a right triangle to the lengths of its sides. The primary trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).

What is the unit circle and its relation to trigonometric functions?

Unit circle is a circle with a radius of 1 unit centered at the origin of a Cartesian coordinate system. The trigonometric functions can be defined geometrically using the unit circle, where the coordinates of points on the circle correspond to the values of the trigonometric functions for specific angles.

What are the domains and ranges of trigonometric functions?

Domain of trigonometric functions is all real numbers, while the range depends on the specific function. For sine and cosine, the range is [-1, 1], for tangent and cotangent, the range is all real numbers, and for secant and cosecant, the range excludes values where the functions are undefined.

What are the properties of trigonometric functions?

Trigonometric functions have various properties, including periodicity, symmetry, evenness or oddness, amplitude, phase shift, and relationships between complementary angles. These properties are used to simplify expressions, solve equations, and analyze trigonometric functions.

How are trigonometric functions used in geometry and trigonometry?

Trigonometric functions are used to solve problems involving angles and sides of triangles, as well as to analyze periodic phenomena such as oscillations, waves, and circular motion. They are fundamental in fields such as navigation, engineering, physics, and astronomy.

What are inverse trigonometric functions?

Inverse trigonometric functions, denoted as arcsin, arccos, arctan, arccsc, arcsec, and arccot, are functions that “undo” the action of the corresponding trigonometric functions. They return the angle whose trigonometric function value is a given number.

How do trigonometric functions relate to complex numbers?

Trigonometric functions can be extended to complex numbers using Euler’s formula, which relates trigonometric functions to exponential functions. This extension allows for the representation and analysis of periodic phenomena in complex numbers.

What are some common identities and formulas involving trigonometric functions?

Common identities include Pythagorean identities, sum and difference identities, double angle identities, half angle identities, and product-to-sum and sum-to-product identities. These formulas are used to simplify expressions and solve trigonometric equations.

How can I apply trigonometric functions in real-life situations?

Trigonometric functions are used in real-life situations such as measuring heights and distances, analyzing mechanical systems, designing structures, modeling natural phenomena, and solving navigation and surveying problems.

Where can I find resources to learn more about trigonometric functions?

Resources for learning trigonometric functions include textbooks, online courses, instructional videos, interactive tutorials, problem-solving books, and practice exercises available in libraries, educational websites, and online learning platforms.