Tan theta + tan 2 theta + root 3 tan 2 theta tan theta equals to root 3 find the general solution
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Given: tan Ф + tan 2Ф + √3 x tan Ф x tan 2Ф = √3
To find: The general solution?
Solution:
- Now we have given the trigonometric equation as:
tan Ф + tan 2Ф + √3 x tan Ф x tan 2Ф = √3
- Now simplifying it, we get:
tan Ф + tan 2Ф = √3 - √3 x tan Ф x tan 2Ф
tan Ф + tan 2Ф = √3 ( 1 - tan Ф x tan 2Ф )
tan Ф + tan 2Ф / 1 - tan Ф x tan 2Ф = √3
- Now we know that tan( a+ b) = tan a + tan b / 1 - tan a tan b, so:
tan ( Ф + 2Ф ) = √3
tan 3Ф = √3
tan 3Ф = tan π/3
3Ф = π/3
Ф = π/9 or nπ/3 + π/9
Ф = π/9 ( 3n + 1 ) n ∈ Z
Answer:
So the general solution of tan Ф + tan 2Ф + √3 x tan Ф x tan 2Ф = √3 is π/9 ( 3n + 1 ).
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