IIT JEE Main Maths -Unit 14- Trigonometric identities and equations- Study Notes-New Syllabus

IIT JEE Main Maths -Unit 14- Trigonometric identities and equations – Study Notes – New syllabus

IIT JEE Main Maths -Unit 14- Trigonometric identities and equations – Study Notes -IIT JEE Main Maths – per latest Syllabus.

Key Concepts:

  • Trigonometric identities
  • Trigonometric  equations

IIT JEE Main Maths -Study Notes – All Topics

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all permissible values of the variables. These identities are essential tools for simplifying expressions, solving equations, proving relationships, and solving JEE-level problems.

Fundamental Pythagorean Identities

  • \( \sin^2\theta + \cos^2\theta = 1 \)
  • \( 1 + \tan^2\theta = \sec^2\theta \)
  • \( 1 + \cot^2\theta = \csc^2\theta \)

Reciprocal Identities

  • \( \sin\theta = \dfrac{1}{\csc\theta} \)
  • \( \cos\theta = \dfrac{1}{\sec\theta} \)
  • \( \tan\theta = \dfrac{1}{\cot\theta} \)

 Quotient Identities

  • \( \tan\theta = \dfrac{\sin\theta}{\cos\theta} \)
  • \( \cot\theta = \dfrac{\cos\theta}{\sin\theta} \)

 Angle Sum and Difference Identities

For any angles \( A \) and \( B \):

  • \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
  • \( \sin(A – B) = \sin A \cos B – \cos A \sin B \)
  • \( \cos(A + B) = \cos A \cos B – \sin A \sin B \)
  • \( \cos(A – B) = \cos A \cos B + \sin A \sin B \)
  • \( \tan(A + B) = \dfrac{\tan A + \tan B}{1 – \tan A \tan B} \)
  • \( \tan(A – B) = \dfrac{\tan A – \tan B}{1 + \tan A \tan B} \)

Double Angle Identities

  • \( \sin 2\theta = 2\sin\theta \cos\theta \)
  • \( \cos 2\theta = \cos^2\theta – \sin^2\theta \)
  • \( \cos 2\theta = 2\cos^2\theta – 1 \)
  • \( \cos 2\theta = 1 – 2\sin^2\theta \)
  • \( \tan 2\theta = \dfrac{2\tan\theta}{1 – \tan^2\theta} \)

Triple Angle Identities

  • \( \sin 3\theta = 3\sin\theta – 4\sin^3\theta \)
  • \( \cos 3\theta = 4\cos^3\theta – 3\cos\theta \)
  • \( \tan 3\theta = \dfrac{3\tan\theta – \tan^3\theta}{1 – 3\tan^2\theta} \)

Product to Sum Identities

  • \( \sin A \sin B = \dfrac{1}{2}[\cos(A – B) – \cos(A + B)] \)
  • \( \cos A \cos B = \dfrac{1}{2}[\cos(A – B) + \cos(A + B)] \)
  • \( \sin A \cos B = \dfrac{1}{2}[\sin(A + B) + \sin(A – B)] \)

 Sum to Product Identities

  • \( \sin A + \sin B = 2\sin\dfrac{A+B}{2}\cos\dfrac{A-B}{2} \)
  • \( \sin A – \sin B = 2\cos\dfrac{A+B}{2}\sin\dfrac{A-B}{2} \)
  • \( \cos A + \cos B = 2\cos\dfrac{A+B}{2}\cos\dfrac{A-B}{2} \)
  • \( \cos A – \cos B = -2\sin\dfrac{A+B}{2}\sin\dfrac{A-B}{2} \)

Important Identity for JEE

\( \sin^4\theta + \cos^4\theta = 1 – 2\sin^2\theta \cos^2\theta \)

\( \sin A + \sin B + \sin C = 4\sin\dfrac{A}{2}\sin\dfrac{B}{2}\sin\dfrac{C}{2} \) if \( A + B + C = 180^\circ \)

Example 

Simplify \( \sin 60^\circ \cos 30^\circ – \sin 30^\circ \cos 60^\circ \).

▶️ Answer / Explanation

Use identity:

\( \sin A \cos B – \cos A \sin B = \sin(A – B) \)

So expression becomes

\( \sin(60^\circ – 30^\circ) = \sin 30^\circ = \dfrac{1}{2} \)

Example 

Find the value of \( \cos 75^\circ \).

▶️ Answer / Explanation

Write using angle difference:

\( 75^\circ = 45^\circ + 30^\circ \)

\( \cos(45^\circ + 30^\circ) = \cos45^\circ\cos30^\circ – \sin45^\circ\sin30^\circ \)

\( = \dfrac{1}{\sqrt{2}}\cdot\dfrac{\sqrt{3}}{2} – \dfrac{1}{\sqrt{2}}\cdot\dfrac{1}{2} \)

\( = \dfrac{\sqrt{3} – 1}{2\sqrt{2}} \)

Example

If \( \sin\theta + \cos\theta = \sqrt{2}\cos\theta \), find \( \theta \).

▶️ Answer / Explanation

Rewrite equation:

\( \sin\theta + \cos\theta = \sqrt{2}\cos\theta \)

\( \sin\theta = (\sqrt{2} – 1)\cos\theta \)

Divide both sides by \( \cos\theta \):

\( \tan\theta = \sqrt{2} – 1 \)

Angle whose tan is \( \sqrt{2} – 1 \):

\( \theta = 15^\circ \)

Trigonometric Equations

Trigonometric equations involve trigonometric functions like \( \sin\theta \), \( \cos\theta \), \( \tan\theta \), etc. Solving them means finding all angles that satisfy the equation. JEE questions often use identities, transformations, and periodicity.

General Solutions of Basic Equations

(a) \( \sin\theta = k \)

  • If \( -1 \le k \le 1 \): \( \theta = n\pi + (-1)^n \sin^{-1}(k) \)
  • If \( k \notin [-1,1] \): No solution.

(b) \( \cos\theta = k \)

  • If \( -1 \le k \le 1 \): \( \theta = 2n\pi \pm \cos^{-1}(k) \)
  • If \( k \notin [-1,1] \): No solution.

(c) \( \tan\theta = k \)

\( \theta = n\pi + \tan^{-1}(k) \)

Periodicity of Trigonometric Functions

  • \( \sin\theta \) and \( \cos\theta \) period: \( 2\pi \)
  • \( \tan\theta \) and \( \cot\theta \) period: \( \pi \)

This helps derive infinite solutions.

 Important Algebraic Forms

(a) Linear form: \( a\sin\theta + b\cos\theta = c \)

Rewrite using the identity:

\( a\sin\theta + b\cos\theta = \sqrt{a^2 + b^2}\sin(\theta + \alpha) \)

where \( \sin\alpha = \dfrac{b}{\sqrt{a^2 + b^2}} \), \( \cos\alpha = \dfrac{a}{\sqrt{a^2 + b^2}} \).

(b) Quadratic in trigonometric functions

Solve quadratics like \( a\sin^2\theta + b\sin\theta + c = 0 \) or \( a\tan^2\theta + b\tan\theta + c = 0 \).

Then apply general solution of each root.

Key Identity Used in JEE

If \( \sin\theta = \sin\alpha \), then \( \theta = n\pi + (-1)^n\alpha \)

If \( \cos\theta = \cos\alpha \), then \( \theta = 2n\pi \pm\alpha \)

If \( \tan\theta = \tan\alpha \), then \( \theta = n\pi + \alpha \)

 Constraints

  • Consider domain restrictions like \( \tan\theta \ne \infty \).
  • Check for extraneous solutions when multiplying or squaring.

Example

Solve: \( \sin\theta = \dfrac{1}{2} \)

▶️ Answer / Explanation

Angles for which \( \sin\theta = \dfrac{1}{2} \) are \( \theta = 30^\circ \) and \( 150^\circ \).

General solution:

\( \theta = n\pi + (-1)^n\dfrac{\pi}{6} \)

Example 

Solve: \( 3\sin\theta + 4\cos\theta = 2 \)

▶️ Answer / Explanation

Rewrite in form:

\( 3\sin\theta + 4\cos\theta = 5\sin(\theta + \alpha) \)

Where \( \sin\alpha = \dfrac{4}{5},\quad \cos\alpha = \dfrac{3}{5} \).

Equation becomes: \( 5\sin(\theta + \alpha) = 2 \)

\( \sin(\theta + \alpha) = \dfrac{2}{5} \)

General solution:

\( \theta + \alpha = n\pi + (-1)^n\sin^{-1}\left(\dfrac{2}{5}\right) \)

\( \theta = n\pi + (-1)^n\sin^{-1}\left(\dfrac{2}{5}\right) – \alpha \)

Example 

Solve the equation \( \sin\theta + \sin2\theta + \sin3\theta = 0 \).

▶️ Answer / Explanation

Use identity:

\( \sin2\theta = 2\sin\theta\cos\theta \), \( \sin3\theta = 3\sin\theta – 4\sin^3\theta \)

Equation becomes:

\( \sin\theta + 2\sin\theta\cos\theta + 3\sin\theta – 4\sin^3\theta = 0 \)

\( \sin\theta(4 + 2\cos\theta – 4\sin^2\theta) = 0 \)

Case 1: \( \sin\theta = 0 \)

\( \theta = n\pi \)

Case 2: \( 4 + 2\cos\theta – 4\sin^2\theta = 0 \)

Use identity \( \sin^2\theta = 1 – \cos^2\theta \):

\( 4 + 2\cos\theta – 4(1 – \cos^2\theta) = 0 \)

\( 4 + 2\cos\theta – 4 + 4\cos^2\theta = 0 \)

\( 4\cos^2\theta + 2\cos\theta = 0 \)

\( 2\cos\theta(2\cos\theta + 1) = 0 \)

Subcases:

  • \( \cos\theta = 0 \Rightarrow \theta = \dfrac{\pi}{2} + n\pi \)
  • \( 2\cos\theta + 1 = 0 \Rightarrow \cos\theta = -\dfrac{1}{2} \Rightarrow \theta = \dfrac{2\pi}{3} + 2n\pi, \ \dfrac{4\pi}{3} + 2n\pi \)

Complete Solution Set:

  • \( \theta = n\pi \)
  • \( \theta = \dfrac{\pi}{2} + n\pi \)
  • \( \theta = \dfrac{2\pi}{3} + 2n\pi \)
  • \( \theta = \dfrac{4\pi}{3} + 2n\pi \)

Notes and Study Materials

Examples and Exercise

IIT JEE (Main) Mathematics ,”Trigonometrical identities and equations.” Notes ,Test Papers, Sample Papers, Past Years Papers , NCERT , S. L. Loney and Hall & Knight Solutions and Help from Ex- IITian

About this unit

Trigonometrical identities and equations.

IITian Academy Notes for IIT JEE (Main) Mathematics – Trigonometrical identities and equations.

The success mantra of the JEE is practice and hard work. Gone are the days when students used to spend hours in attempting one question. Now it is an era of multiple choice questions. The JEE Mathematics questions test a student’s acquired knowledge as well as his aptitude. We have excellent notes prepared by Ex-IITian to best match the requirement of the exam. Focus is given on problem solving skills and small tips and tricks to do it faster and easier. We , Ex-IITian at https://www.iitianacademy.com. will make sure you understand the concept well.

IIT JEE (Main) Mathematics, Trigonometrical identities and equations. Solved Examples and Practice Papers.

Get excellent practice papers and Solved examples to grasp the concept and check for speed and make you ready for big day. These Question Papers are prepared by Ex-IITIan for IIT JEE (Main) Mathematics , Trigonometrical identities and equations..

S. L. Loney IIT JEE (Main) Mathematics

This book is the one of the most beautifully written book by the author. Trigonometry is considered to be one of the easiest topics in mathematics by the aspirants of IIT JEE, AIEEE and other state level engineering examination preparation. It would not be untrue to say that most of the sources have taken inspiration from this book as it is the most reliable source. The best part of this book is its coverage in Heights and Distances and Inverse Trigonometric Functions. The book gives a very good learning experience and the exercises which follow are not only comprehensive but they have both basic and standard questions.. I will help you online for any doubt / clarification.

Hall & Knight IIT JEE (Main) Mathematics

Algebra by Hall and Knight is one of the best books for JEE preparation. Students preparing for IIT JEE and other engineering entrance exams as well as students appearing for board exams should read this everyday, especially to master Algebra and Trigonometrical identities and equations.. Hall and Knight have explained the concepts logically in their book.

IIT JEE (Main) Mathematics Assignments

Chapter wise assignments are being given by teachers to students to make them understand the chapter concepts. Its extremely critical for all CBSE students to practice all assignments which will help them in gaining better marks in examinations. All assignments available for free download on the website are developed by the best teachers having many years of teaching experience in CBSE schools all over the country. Students, teachers and parents are advised to contact me online incase of any doubt / clarification.

Past Many Years (40 Years) Questions IIT JEE (Main) Mathematics Solutions Trigonometrical identities and equations.

Past 40 Years Question Papers Solutions for IIT JEE (Main) Mathematics Trigonometrical identities and equations. are provided here with simple step-by-step explanations. These solutions for Trigonometrical identities and equations. are extremely popular among IIT JEE (Main) students for Chemistry . Trigonometrical identities and equations. Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Past Many Years Question Papers Book of IIT JEE (Main) Mathematics Chapter Trigonometrical identities and equations. are provided here for . I will help you online for any doubt / clarification.

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