The solution of equation 5 sin^2 x - 7 sin x cos x + 16 cos^2 x - 4 is
The solution of equation 5 sin2 x - 7 sin x cos x + 16 cos2 x - 4 is
(1) x = nπ + tan-1 3 or x = nπ + tan-1 4
(2) x = nπ + π/6 or x = nπ + π/4
(3) x = nπ or x = nπ + π/4
(4) None of these
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Correct option (1) x = nπ + tan-1 3 or x = nπ + tan-1 4
Explanation :
To solve the kind of equation; we use the fundamental formula trigonometrical identity,
sin2 x + cos2 x = 1 writing the equation in the form,
5 sin2 x – 7 sin x cos x +16 cos2 x = 4 (sin2 x + cos2 x)
= sin2 x – 7 sin x cos x + 12 cos2 x = 0
Dividing by cos2 x on both sides we get,
tan2 x –7 tan x + 12 = 0
Now it can be factorized as ;
(tan x – 3) (tan x – 4) =0
tan x = 3, 4
i.e. tan x = tan (tan–13)
or tan x = tan–1 (tan 4)
x = nπ + tan-1 3 or x = nπ + tan-1 4
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