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INVERSE TRIGONOMETRIC FUNCTIONS in 1 shot - All Concepts, Tricks & PYQs Covered | JEE Main & Adv.

JEE Wallah・168 minutes read

An emphasis on understanding trigonometry functions, particularly inverse functions like sine inverse x, highlights the importance of mastering the concepts thoroughly, with an estimated completion time of 4 to 5 hours. The properties and characteristics of inverse trigonometric functions are discussed, focusing on domains, ranges, graphs, and relationships between sine, cosine, secant, and cosecant inverses for effective problem-solving.

Insights

Understanding the domain and range of inverse trigonometric functions is crucial for effective problem-solving.
Thoroughly learning the basics of trigonometry functions is emphasized for future benefits.
The graphs and properties of inverse trigonometric functions like sine inverse x and cosine inverse x are detailed to showcase their unique characteristics.
The relationship between x and a in trigonometric functions like tan inverse x + tan inverse a is explored through practical examples.
Inequality reversal and the impact of inverse functions on equations are highlighted, emphasizing the need for verification of solutions in mathematical problem-solving.

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Recent questions

What are inverse trigonometric functions?
Inverse trigonometric functions are functions that reverse the output of trigonometric functions, allowing us to find angles based on given values.

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Summary

00:00

Mastering Inverse Trigonometry Functions in Hours

Trigonometry functions are discussed, focusing on basic concepts and properties.
The importance of understanding the domain and range of inverse trigonometric functions is highlighted.
The chapter emphasizes learning the basics thoroughly for future benefits.
The duration to complete the chapter is estimated to be around 4 to 5 hours.
Stress is placed on understanding the concepts to solve questions effectively.
Inverse trigonometric functions are noted for their conceptual weightage in questions.
The session begins with the introduction of inverse trigonometric functions like sine inverse x and cosine inverse x.
The range and domain of sine inverse x are detailed, emphasizing the input-output relationship.
The graph of sine inverse x is explained, showcasing its symmetry with the sine x graph.
The increasing nature and bounded range of sine inverse x are highlighted, along with its odd function characteristics.

15:18

"Inverse Cosine Function: Properties and Graph"

The graph should be placed before zero, specifically between zero and pi, with a domain of -π/2 to π/2.
The function A = cos x is defined as having an inverse function, cos inverse x.
The range of cos inverse x is between zero and pi, with the graph lying solely on the X axis.
The inverse of cos x, cos inverse x, has a range from -π/2 to π/2.
The value of cos 0 is zero, while cos inverse 0 is π/2.
The graph of A = cos⁻¹ x is a mirror image of the graph of cos x.
The function cos inverse x is a decreasing function, with a range from -π/2 to π/2.
Cos inverse x is an odd function, as it throws out negative values.
The value of cos inverse -1/2 is -π/6, while cos inverse 1/2 is π/6.
The property of cos inverse -x is that it equals π - cos inverse x, giving angles in the second quadrant.

29:14

Inverse Trig Functions: Properties and Graphs

The function is bounded between -π/2 and π/2, with a range from -π/2 to π/2.
As x approaches infinity, the angle approaches π/2, making tan⁻¹ infinity equal to π/2.
The graph of tan⁻¹ remains zero, even as x approaches infinity.
Tan⁻¹ is an increasing function, similar to sin⁻¹, and is an odd function.
The value of tan⁻¹ remains zero, and as x approaches infinity, it approaches π/2.
The graph of a = 10x is symmetric and monotonically increasing.
The graph of cos⁻¹x is between 0 and π, with a value of π/2 at x = π/2.
The graph of a = cos⁻¹x is continuous and non-periodic, with a range between 0 and π.
The graph of sin⁻¹x is between -π/2 and π/2, with a value of π/2 at x = 0.
Sin⁻¹x, cos⁻¹x, and tan⁻¹x are all odd functions, providing angles in the first and fourth quadrants based on the sign of x.

46:55

Trigonometric Functions in Third Quadrant

In the third quadrant, the angle for 2π/3 is not used.
The inverse trigonometric function in the third quadrant is discussed.
The value of √2 is determined to be -π/4.
The process of finding the value of inverse - 1 / √2 from the second quadrant is explained.
The angle for cos inverse with a negative value is calculated to be 3π/4.
The concept of throwing out negative values for inverse tan is highlighted.
The angle for tan inverse of -√3 is found to be -π/2.
The proof of a trigonometric function involving sin inverse 3/5 is demonstrated.
The domain and range of cos inverse of the greatest integer of x are discussed.
The range of sine inverse of A^x is explained to be between 0 and π/2.

01:03:57

Properties of Reciprocals and Inequalities

The expression 1/x + x/x can be written as cosec inverse.
When input is given, 1/x + 1 should be greater than or equal to 0.
If x is positive, then 1/x should be greater than or equal to zero.
The reciprocal of a positive number is positive.
If x is positive, its reciprocal must also be positive.
If 1/x < -2, then x should be greater than -1/2.
The inequality changes when reciprocated.
If 1 + 2x/x <= 0, then 1/x + 2 should be less than or equal to zero.
The critical point of the denominator is zero.
The sign of sine inverse x is always equal to x.

01:19:42

"Graphs, Equations, and Inverse Sine Functions"

Mountains are formed between zero and pi on the x-axis.
The graph is cut at 2π and 4π on the x-axis.
The equation of the line is A = x - 2π.
The slope of the line is positive with a value of 1.
The equation of the line is A = -x + C.
The value of C is -2π.
Sin inverse of sin 5π/3 is π/3.
Sin inverse of sin π/9 is -π/9.
Sin inverse of sin 17π/9 is -π/9.
The graph of sin inverse sin x forms mountains between zero and pi/2, cutting at integral multiples of π.

01:35:56

"Energy Loss, Inverse Trigonometric Functions Explained"

Concentrating on one place leads to energy loss
Energy drinks do not increase energy
The graph of energy increase resembles a specific pattern
Values of cos inverse are discussed
Inverse of cos 3 equals 3
Cosine is the inverse of cosine x
Calculations involving cos inverse of various values are detailed
Graphs of sin⁻¹ sin x and cos⁻¹ cos x are discussed
The graph of tan x and its inverse are explained
Properties of inverse trigonometric functions are outlined, including relationships between sine, cosine, secant, and cosecant inverses

01:54:23

Trigonometric Identities and Properties Explained

Pi/2 is the value of sin inverse - 1/2 + cos inverse - 1/2.
The value of cos inverse - 1/2 is equal to -π/6.
The value of cos inverse - 1/2 is equal to -π/3.
The sum of π - π/6 - π/3 is equal to π/2.
The property states that sin inverse x + tan inverse 1/x = π/2.
For tan inverse x + tan inverse 1/x = π/2, x must be positive.
The value of tan inverse -√3 + tan inverse - 1/√3 is -π/3.
The value of tan inverse -√3 + tan inverse - 1/√3 is -π/2.
The maximum and minimum values of sin inverse x + cos inverse x are π/32 and 428π/32, respectively.
The range of sin inverse x + cos inverse x is between 1/32 and 75³.

02:13:15

Optimal Values in Trigonometric Functions

When given π/4, the time of π/4 plus the time of π/4 will become what plus π/4.
Pi by 32 becomes its minimum value when x equals -1.
The maximum value is found when k is equal to a.
The maximum value of x is found when sin inverse x equals 5/2.
The maximum value is obtained when x equals -1.
The minimum value is found when x equals 0.
The minimum value is obtained when x equals van by root tu.
The maximum value is achieved when x equals 1/√2.
The minimum value is π/4, and the maximum value is 3π/4.
The minimum value is π/4, and the maximum value is 7/8.

02:54:54

"Formulas for tan inverse x + tan inverse a"

Two different formulas for tan inverse x + tan⁻¹ a are presented.
The angles alpha and beta are introduced, with x and a being positive numbers.
The relationship between x and a in terms of theta is explained.
The equation tan(α + β) = tan α + tan β is derived.
The value of tan inverse x + tan inverse a is determined based on the value of theta.
Two cases are considered based on the range of theta: 0 to π/2 and π/2 to π.
The graph of tan inverse x is discussed, highlighting the parallel lines formed.
The formula for tan inverse x - tan⁻¹ a is presented.
Practical examples are provided to illustrate the application of the formulas.
The conversion of trigonometric functions to different forms is demonstrated through examples involving secant and sine functions.

03:10:53

Solving Math Problems with Trigonometric Formulas

The calculation involves various mathematical operations like subtraction, multiplication, and division.
The formula for tan inverse is applied to simplify the calculations.
The process includes taking the LCM, canceling out common factors, and simplifying fractions.
The formula for tan inverse is applied to different sets of numbers.
The calculations involve finding common factors, simplifying fractions, and applying trigonometric formulas.
The process includes converting sine inverse to terms of tan inverse.
The calculations involve Pythagorean triplets and trigonometric functions.
The final answer is determined by converting sine inverse to cosine inverse.
The process involves creating right-angle triangles and calculating the values of sides.
The final answer is obtained by applying trigonometric formulas and simplifying the expressions.

03:28:57

Mathematical Operations and Series Simplification

Pythagorean theorem: If perpendicular is 63 and base is 16, the hypotenuse will be 65.
Unit digits: The unit digit of 16² is 6, while the unit digit of 63² is 9.
Trigonometric calculations: Using inverse trigonometric functions to find angles and values.
Simplification: Simplifying inverse trigonometric functions using specific formulas.
Graphs: Understanding the graphs of inverse trigonometric functions.
Standard simplifications: Exploring standard simplifications of inverse trigonometric functions.
Series questions: Breaking down series questions into manageable parts.
Factorization: Identifying and utilizing factorization in mathematical equations.
Infinite series: Solving infinite series questions by substituting values and simplifying.
Final answer: Summing up the series to find the final answer.

03:49:57

Self-improvement before relationships: A mathematical guide

Encouragement to improve oneself through reading, writing, and self-improvement
Emphasis on being the best version of oneself before entering a relationship
Advice on respecting oneself and others in relationships
Introduction to mathematical problems involving inverse functions
Explanation of the process to convert standard series into inverse terms
Detailed steps on calculating inverse values using trigonometric functions
Instructions on breaking down complex mathematical expressions
Clarification on the characteristics of increasing and decreasing functions
Practical example of solving inequalities involving inverse trigonometric functions
Analysis of factors and critical points in solving mathematical equations

04:09:58

Inverse Functions and Inequality Verification

Inequality reversal: Negative values are sent to the pay division, causing the inequality to be reversed. Inverse X is reduced by 2, resulting in a value less than zero.
Inverse functions and inequality: The inverse function of x - 2 and x - 3 are considered, with their product being greater than zero due to the nature of inverse functions.
Solving equations: Equations involving inverse trigonometric functions are solved, leading to specific values for x such as 0, 1/2, and -1/2.
Verification of solutions: The obtained solutions are verified to ensure they satisfy the equations, emphasizing the importance of confirming results in mathematical problem-solving.

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