If \(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}} = \text{cosec}\theta\), then what is \(\sin\theta\) equal to?
Evaluating the Infinite Nested Radical
The problem asks us to find the value of \(\sin\theta\) given the equation \(\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}} = \text{cosec}\theta\). First, let's evaluate the infinite nested radical expression. Let the value of the expression be \(x\). \(x = \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+...}}}}\) Notice that the expression under the square root contains the original expression itself. We can rewrite this relationship as:
\(x = \sqrt{2+x}\)To solve for \(x\), we can square both sides of the equation:
\(x^2 = (\sqrt{2+x})^2\) \(x^2 = 2+x\)Now, rearrange this into a standard quadratic equation form (\(ax^2 + bx + c = 0\)):
\(x^2 - x - 2 = 0\)We can solve this quadratic equation by factoring:
\((x-2)(x+1) = 0\)This gives us two possible solutions for \(x\): \(x=2\) or \(x=-1\). Since \(x\) represents the value of a square root (specifically, the principal square root), it must be a non-negative value. Therefore, we must choose \(x=2\). So, the value of the infinite nested radical is 2.
Relating Radical Value to Trigonometric Function
The problem states that the value of the nested radical is equal to \(\text{cosec}\theta\). We found the value of the radical to be 2. Therefore:
\(\text{cosec}\theta = 2\)Calculating sin(theta)
We need to find the value of \(\sin\theta\). We know the fundamental relationship between \(\sin\theta\) and \(\text{cosec}\theta\):
\(\sin\theta = \frac{1}{\text{cosec}\theta}\)Substitute the value of \(\text{cosec}\theta\) we found:
\(\sin\theta = \frac{1}{2}\)Thus, the value of \(\sin\theta\) is \(\frac{1}{2}\).
Similar Questions
- If \(2 \tan\theta = \sec^2\theta-2\), where \(0 < \theta < \pi /2\), then what is \(\cot\theta\) equal to?
- What is \(\frac{\tan^2 \alpha}{\tan^2 \beta}\) equal to?
- If \(\alpha\) and \(\beta\) are complementary angles, then which one of the following is correct?
- What is \(\sec\theta\) equal to?
- What is \(\tan\theta\) equal to?
- What is the relation between \(p\) and \(q\)?
- What is $\tan\theta + \cot\theta$ equal to?
- Consider the following statements :
1. In a triangle ABC, if \(\sin A + \sin B + \sin C = \frac{3\sqrt{3}}{2}\), then the triangle can be equilateral.
2. In a triangle ABC, if \(\cos A + \cos B + \cos C = \frac{3}{2}\), then the triangle can be equilateral.
Which of the statements given above is/are correct? What is \((1 + \tan\alpha \tan\beta)^2 + (\tan\alpha - \tan\beta)^2\) equal to ?
Consider the following statements :
I. \(\tan 50^\circ - \cot 50^\circ\) is positive
II. \(\cot 25^\circ - \tan 25^\circ\) is negative
Which of the statements is/are correct?
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