Let \(\begin{array}{l} S=\left[-\pi, \frac{\pi}{2}\right)-\left[-\frac{\pi}{2},-\frac{\pi}{4},-\frac{3\pi}{4},\frac{\pi}{4}\right].\end{array}\)Then the number of elements in the set \(\begin{array}{l} A=\left\{\theta\in S:\tan\theta\left(1+\sqrt{5}\tan\left(2\theta\right)\right)=\sqrt{5}-\tan\left(2\theta\right) \right\} \end{array}\)is ________.
Solution and Explanation
Here, \(\begin{array}{l} tan~\alpha =\sqrt{5}\end{array}\)
\(\begin{array}{l} \therefore\ \tan\theta=\frac{\tan\alpha-\tan2\theta}{1+\tan\alpha\tan2\theta} \end{array}\)
∴ tan θ = tan (α – 2θ)
\(α – 2θ = nπ + θ\)
⇒ \(3θ = α – nπ\)
\(\begin{array}{l} \Rightarrow\ \theta = \frac{\alpha}{3}-\frac{n\pi}{3}~;~n\in Z\end{array}\)
If θ [–π, π/2) then
n = 0, 1, 2, 3, 4 are acceptable
∴ 5 solutions.
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".