Problem 34 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following inequalities: $$ \left(\cot ^{-1} x\right)^{2}-5\left(\cot ^{-1} x\right)+6>0 $$

Step-by-Step Solution

Answer

The solution to the inequality \((\cot^{-1} x)^2 -5(\cot^{-1} x)+6 > 0\) is cot(3) > x > cot(2).

Step 1: Simplify using Substitution

First, let's simplify the equation to make it easier to solve by setting \(\cot^{-1} x\) equal to 'a'. \n So, the inequality becomes: \(a^2 - 5a + 6 > 0\)

Step 2: Solve Inequality with Factoring

Next, we factor the quadratic inequality into: \((a-2)(a-3) > 0\)

Step 3: Find Solution Set

For the solution set of the inequality \((a-2)(a-3) > 0\), consider the open intervals (-∞, 2), (2, 3), and (3, +∞). We find the test points for each interval: 0, 2.5, and 4. Substituting these points into the inequality, we can determine which intervals are part of the solution set. \n - When a = 0, the inequality (0 - 2) * (0 - 3) > 0 is not satisfied. - When a = 2.5, the inequality (2.5 - 2) * (2.5 - 3) > 0 is satisfied.- When a = 4, the inequality (4 - 2) * (4 - 3) > 0 is not satisfied.\nSo the solution set for 'a' is (2, 3)

Step 4: Substitute Back Original Variable

Now we substitute \(\cot^{-1} x\) back in for 'a' in the solution set to find the range of the original variable 'x'. \nSo the solution is \(\cot^{-1} x\) in (2, 3), or cot(3) > x > cot(2)

Key Concepts

Understanding Cotangent

The cotangent is a fundamental trigonometric function. To put it simply, cotangent, denoted as \( \cot \), is the reciprocal of tangent. So, \( \cot(\theta) = \frac{1}{\tan(\theta)} \).

In right-angled triangles, it is defined as the ratio of the adjacent side to the opposite side of a given angle. It's essential in various mathematical problems, especially when dealing with circle-related calculations and angles.

Cotangent behaves differently in different quadrants of the unit circle, swapping between positive and negative values. This characteristic is crucial when solving trigonometric inequalities, as it influences the sign of our solutions.

• When \( \cot(\theta) > 0 \): \( \theta \) is in Quadrants I and III.
• When \( \cot(\theta) < 0 \): \( \theta \) is in Quadrants II and IV.

Understanding cotangent can help in solving inequalities like the one given in the exercise when applying transformations and solving for variables.

Inverse Trigonometric Functions

Inverse trigonometric functions, as their name implies, "undo" the regular trigonometric functions, providing an angle as a result. For cotangent, the inverse is denoted as \( \cot^{-1} x \) or arccotangent. It finds the angle whose cotangent is \( x \).

Working with inverse trigonometric functions can sometimes be tricky due to their restricted range to maintain the function's validity and prevent repeating values.

• For \( \cot^{-1} x \), the principal range is generally \( (0, \pi) \), ensuring a unique output angle.
• Application of \( \cot^{-1} \) often results in angles that might not conventionally appear when using direct trigonometric functions.

Comprehending the behavior of inverse functions is vital when back-substituting in a problem, as incorrect handling can cause errors in the solution.

Solving Quadratic Inequalities

Quadratic inequalities involve inequalities with a squared variable. In our problem, the expression \( a^2 - 5a + 6 > 0 \) is a quadratic inequality that arises from setting the inverse cotangent substitution.

To solve it, the quadratic expression must be factored, allowing us to find intervals where the inequality holds true.

• Step 1: Rewrite the inequality in standard form. Here it's already provided.
• Step 2: Factorize the quadratic expression. Our factorization results in \( (a-2)(a-3) > 0 \).
• Step 3: Determine test intervals. Use the roots, here 2 and 3, to identify intervals.
• Step 4: Select test points from each interval: choose 0, 2.5, and 4 to evaluate the inequality.
• Step 5: Identify where the inequality holds. In this example, the solution is in the interval \( (2,3) \).

Such methodical handling of quadratic inequalities provides us with the solutions for the variable, leading to conclusions about the original variable or expression analyzed in context with the inequality.