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Chapter 19: Problem 1848

If the roots of the quadratic equation \(4 x^{2}-4 x+1=\cos ^{2} \theta\) is \(a\) and \(\beta\) then \(\alpha+\beta=\) (a) \(\cos ^{2}(\theta / 2)\) (b) \(\sin ^{2}(\theta / 2)\) (c) 1 (d) \(2 \cos ^{2}(\theta / 2)\)

Short Answer

Expert verified
(c) 1

Step by step solution

01

Write down the given quadratic equation and compare it with standard form

The given quadratic equation is: \[4x^2 - 4x + 1 = \cos^2{\theta}\] We'll rewrite it as: \[4x^2 - 4x + (1 - \cos^2{\theta}) = 0\] Now, this quadratic equation is in the standard form \(ax^2 + bx + c = 0\), where \(a = 4\), \(b = -4\), and \(c = 1 - \cos^2{\theta}\).
02

Use the sum of the roots formula

Now that we have the coefficients of the quadratic equation, we can apply the sum of the roots formula: \[\alpha + \beta = -\frac{b}{a}\] Plugging the values of a and b: \[\alpha + \beta = -\frac{-4}{4}\]
03

Calculate the sum of the roots

Now, we just need to calculate the value: \[\alpha + \beta = 1\] This corresponds to option (c). #Answer# (c) 1

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