Math Problem Statement

sin(2 cos^(-1)(2x)) is defined for x ∈ (√5/4, 1/2). It can be shown that sin(2 cos^(-1)(2x)) is equal to a√(1 - b²x²) where a and b are positive constants. The values of a and b are:

Solution

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Question 12

The equation in question is: sin(2cos1(2x))=1b2x2\sin(2 \cos^{-1}(2x)) = \sqrt{1 - b^2 x^2} where aa and bb are positive constants, and the domain of xx is given by x(54,12)x \in \left( \frac{\sqrt{5}}{4}, \frac{1}{2} \right).

We need to determine the values of aa and bb.

Question 13

We are given the function: f(x)=asin1(x)+bf(x) = a \sin^{-1}(x) + b where aa and bb are positive constants, and the range of f(x)f(x) is (23π,3π+2)(2 - 3\pi, 3\pi + 2). We need to find the values of aa and bb.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Inverse Functions
Square Roots

Formulas

sin(2θ) = 2sin(θ)cos(θ)
cos^(-1)(x)
Pythagorean identity

Theorems

Inverse trigonometric functions identity
Pythagorean theorem

Suitable Grade Level

Grades 11-12