An interval in mathematics refers to a set of real numbers between two endpoints. It can be open, closed, or half-open, depending on whether the endpoints are included. Intervals are used to define domains and ranges and they help describe continuous values in functions and inequalities.
Class 12 Maths Chapter 2 Inverse Trigonometric Functions deals with the inverse of trigonometric functions like sine and cosine and tangent or cosecant and secant and cotangent. It explains how to find angles from given values of these functions. The chapter covers domains and ranges graphical representations and solving equations involving inverse trigonometric functions.
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We are given to find the interval where sin⁻¹x > cos⁻¹x.
Step 1: Recall the properties of inverse trigonometric functions The range of sin⁻¹x is [-π/2, π/2] and the range of cos⁻¹x is [0, π].
For the condition sin⁻¹x > cos⁻¹x to be true, the values of x must satisfy:
sin⁻¹x > cos⁻¹x
Step 2: Use the identity sin⁻¹x + cos⁻¹x = π/2 From the identity:
sin⁻¹x + cos⁻¹x = π/2 We can subtract cos⁻¹x from both sides to get:
sin⁻¹x = π/2 – cos⁻¹x Thus, for sin⁻¹x > cos⁻¹x, we need:
π/2 – cos⁻¹x > cos⁻¹x That is:
π/2 > 2cos⁻¹x which gives:
cos⁻¹x 1/√2
Hence, the required condition sin⁻¹x > cos⁻¹x holds when x belongs to:
(1/√2, 1)
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