The number of real values of a satisfying the equation a^(2)-2a sin x+1=0 is

Question

The number of real values of  a satisfying the equation  a^(2)-2a sin x+1=0 is

Bytelearn · Accepted Answer

Given Quadratic Equation: We are given the quadratic equation in terms of a: a^(2) - 2a sin x + 1 = 0 This is a quadratic equation with respect to a, and it can be written in the standard form as: a^(2) - (2 sin x)a + 1 = 0 To determine the number of real values of a that satisfy this equation, we need to consider the discriminant of the quadratic equation, which is given by: D = b^(2) - 4ac Here, a = 1 (coefficient of a^(2)), b = -2 sin x (coefficient of a), and c = 1 (constant term).Calculate Discriminant: Calculate the discriminant using the values of a, b, and c: D = (-2 sin x)^(2) - 4(1)(1) D = 4 sin^(2) x - 4 Since sin^(2) x is always less than or equal to 1 for all real x, the maximum value of 4 sin^(2) x is 4.Analyzing Discriminant: Analyze the discriminant: If D > 0, there are two distinct real values of a. If D = 0, there is exactly one real value of a (a repeated root). If D < 0, there are no real values of a (the roots are complex).Range of Discriminant: Determine the range of D based on the maximum value of sin^(2) x: Since the maximum value of 4 sin^(2) x is 4, the discriminant D can be at most 0 (when sin^(2) x = 1). Therefore, the discriminant D is always less than or equal to 0. D <= 0Concluding Real Values: Conclude the number of real values of a: Since D <= 0, the equation a^(2) - 2a sin x + 1 = 0 will have at most one real value of a. When D = 0 (which occurs when sin^(2) x = 1), there is exactly one real value of a. When D < 0 (which occurs when sin^(2) x < 1), there are no real values of a.

The number of real values of $a$ satisfying the equation $a^{2}-2a \sin x+1=0$ is

Full solution

More problems from Domain and range of quadratic functions: equations

Related topics

The number of real values of aaa satisfying the equation a2−2asin⁡x+1=0a^{2}-2a \sin x+1=0a2−2asinx+1=0 is

The number of real values of $a$ satisfying the equation $a^{2}-2a \sin x+1=0$ is