Inverse Trigonometric Functions

Let $2%20%5Ccos%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B2%7D%7B%5Csqrt%7B5%7D%7D%20%5Cright)%20%3D%20%5Ctheta.$

Step 1: Expressing in Terms of Half-Angle

$%5Cfrac%7B2%7D%7B%5Csqrt%7B5%7D%7D%20%3D%20%5Ccos%20%5Cfrac%7B%5Ctheta%7D%7B2%7D$

Therefore, $%5Ctan%20%5Cfrac%7B%5Ctheta%7D%7B2%7D%20%3D%20%5Cfrac%7B1%7D%7B2%7D.$

Step 2: Calculating Individual Trigonometric Functions

$%5Ctan%20%5Csin%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B3%7D%7B5%7D%20%5Cright)%20%3D%20%5Cfrac%7B3%7D%7B4%7D%2C%20%5Cquad%20%5Ctan%20%5Ccos%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B2%7D%7B%5Csqrt%7B5%7D%7D%20%5Cright)%20%3D%20%5Cfrac%7B4%7D%7B3%7D.$

Step 3: Applying the Tangent Subtraction Formula

$%5Ctan%20%5Cleft(%20%5Csin%5E%7B-1%7D%20%5Cfrac%7B3%7D%7B5%7D%20-%20%5Ccos%5E%7B-1%7D%20%5Cfrac%7B2%7D%7B%5Csqrt%7B5%7D%7D%20%5Cright)%20%3D%20%5Cfrac%7B%5Cfrac%7B3%7D%7B4%7D%20-%20%5Cfrac%7B4%7D%7B3%7D%7D%7B1%20%2B%20%5Cfrac%7B3%7D%7B4%7D%20%5Ccdot%20%5Cfrac%7B4%7D%7B3%7D%7D$

Final Calculation

$%3D%20%5Cfrac%7B%5Cfrac%7B9%7D%7B12%7D%20-%20%5Cfrac%7B16%7D%7B12%7D%7D%7B1%20%2B%20%5Cfrac%7B12%7D%7B12%7D%7D$ $%3D%20%5Cfrac%7B-7%7D%7B24%7D$

Step 1: Let $x%20%3D%20%5Ctan%20%5Ctheta$ . Then $%5Ctheta%20%3D%20%5Ctan%5E%7B-1%7D%20x$ Substitute in the given equation: $%5Ctan%5E%7B-1%7D%20x%20%3D%20%5Ctan%5E%7B-1%7D(2x)%20-%20%5Cfrac%7B1%7D%7B2%7D%20%5Csin%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B6x%7D%7B9%20%2B%20x%5E2%7D%20%5Cright)$
Step 2: Apply $%5Ctan%5E%7B-1%7D%20x%20%3D%20%5Ctheta%20%5CRightarrow%20x%20%3D%20%5Ctan%20%5Ctheta$ and the domain restriction $%5Ctheta%20%5Cin%20(-%5Cfrac%7B%5Cpi%7D%7B2%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D)%20%5CRightarrow%20x%20%5Cin%20%5Cmathbb%7BR%7D$ We rewrite the equation: $%5Ctan%5E%7B-1%7D%20x%20%3D%20%5Ctan%5E%7B-1%7D(2x)%20-%20%5Cfrac%7B1%7D%7B2%7D%20%5Csin%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B6x%7D%7B9%20%2B%20x%5E2%7D%20%5Cright)$
Step 3: Define LHS = RHS and analyze the function: Let $f(x)%20%3D%20%5Ctan%5E%7B-1%7D%20x%20-%20%5Ctan%5E%7B-1%7D(2x)%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%5Csin%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B6x%7D%7B9%20%2B%20x%5E2%7D%20%5Cright)$ We want to solve $f(x)%20%3D%200$
Step 4: Observe that $%5Ctan%5E%7B-1%7D%20x%20-%20%5Ctan%5E%7B-1%7D(2x)%20%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7Bx%20-%202x%7D%7B1%20%2B%202x%5E2%7D%20%5Cright)%20%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B-x%7D%7B1%20%2B%202x%5E2%7D%20%5Cright)$ So, $f(x)%20%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B-x%7D%7B1%20%2B%202x%5E2%7D%20%5Cright)%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%5Csin%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B6x%7D%7B9%20%2B%20x%5E2%7D%20%5Cright)$ Step 5: Let us analyze $f(x)%20%3D%200%20%5CRightarrow$ $%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B-x%7D%7B1%20%2B%202x%5E2%7D%20%5Cright)%20%3D%20-%5Cfrac%7B1%7D%7B2%7D%20%5Csin%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B6x%7D%7B9%20%2B%20x%5E2%7D%20%5Cright)$ So define a function: $g(x)%20%3D%20%5Ctan%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B-x%7D%7B1%20%2B%202x%5E2%7D%20%5Cright)%20%2B%20%5Cfrac%7B1%7D%7B2%7D%20%5Csin%5E%7B-1%7D%20%5Cleft(%20%5Cfrac%7B6x%7D%7B9%20%2B%20x%5E2%7D%20%5Cright)$ We now find the number of real solutions of $g(x)%20%3D%200$
Step 6: Graphical/Monotonicity Analysis (or plotting): Using symmetry and bounds: - $%5Cleft%7C%20%5Cfrac%7B-x%7D%7B1%20%2B%202x%5E2%7D%20%5Cright%7C%20%5Cleq%20%5Cfrac%7B1%7D%7B2%5Csqrt%7B2%7D%7D%20%5CRightarrow%20%5Ctan%5E%7B-1%7D%20%5Ctext%7B%20bounded%7D$ - $%5Cleft%7C%20%5Cfrac%7B6x%7D%7B9%20%2B%20x%5E2%7D%20%5Cright%7C%20%5Cleq%201%20%5CRightarrow%20%5Csin%5E%7B-1%7D%20%5Ctext%7B%20defined%7D$
Graphing $g(x)$ shows that it crosses the x-axis three times. So, the number of real solutions is 3.

About Inverse Trigonometric Functions - JEE-ADVANCED

Inverse Trigonometric Functions is a vital chapter for JEE-ADVANCED aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Inverse Trigonometric Functions PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Inverse Trigonometric Functions carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

About Inverse Trigonometric Functions - JEE-ADVANCED

Frequently Asked Questions

Why focus on Inverse Trigonometric Functions PYQs?

How to best use this analysis?

Inverse Trigonometric Functions

High-Yield Trend

Chapter Questions 4 MCQs

Official Solution

Official Solution

Official Solution

Official Solution

About Inverse Trigonometric Functions - JEE-ADVANCED

Frequently Asked Questions

Why focus on Inverse Trigonometric Functions PYQs?

How to best use this analysis?

Chapter Questions
4 MCQs