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in Trigonometry by (53.1k points)

Find the solution of equation 5 sin2 x - 7 sin x cos x + 16 cos2 x = 4.

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To solve this kind of equation, we use the fundamental formula: 

Trigonometrical identity, sin2x + cos2 x = 1.

Writing the equation in the form

5sin2 x - 7 sin x cos x + 16 cos2x = 4(sin2 x + cos2 x)

and simplifying, we get

sin2x - 7 sinx cos x + 12cos2x = 0

Dividing by cos2 x on both sides, we get

tan2 x - 7tan x + 12 = 0

Now it can be factorized as

(tanx  - 3)(tanx - 4) = 0 ⇒ tan x = 3 or 4

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