Question

7) Find the possible value of
tan x; if cos²x+ 5sinx cos x = 3​

Answer 1

Question:-

➡ Find the possible values of tan(x) if cos²(x) + 5 sin(x)cos(x) = 3

Answer:-

➡ The possible values of tan(x) are 2/3 and 1 i.e.,

tan(x) = 1 or tan(x)=⅔

Step By Step Solution:-

Given equation,

➡ cos²(x) + 5 sin(x) cos(x) = 3

Dividing both sides by cos²(x), we get,

➡ cos²(x)/cos²(x) + 5sin(x)cos(x)/cos²(x) = 3/cos²(x)

➡ 1 + 5sin(x)/cos(x) = 3 sec²(x)

Now, we know that,

sec²(x) = 1 + tan²(x)

So,

➡ 1 + 5tan(x) = 3(1 + tan²(x))

➡ 1 + 5tan(x) = 3 + 3tan²(x)

Now, arranging the terms in decreasing order of power, we get,

➡ 3tan²(x) - 5tan(x) + 2 = 0

Now, we have to solve the equation,

➡ 3tan²(x) - 3tan(x) - 2tan(x) + 2 = 0

➡ 3tan(x)(tan(x) - 1) - 2(tan(x) - 1) = 0

➡ (3tan(x) - 2)(tan(x) - 1) = 0

By zero product rule,

➡ 3tan(x) - 2 = 0 or tan(x) - 1 = 0

So,

3tan(x) - 2 = 0

➡ 3tan(x) = 2

➡ tan(x) = ⅔

Also,

tan(x) - 1 = 0

➡ tan(x) = 1

Hence, the possible values of tan(x) are ⅔ and 1.

Note:-

➡ sec²(x) = 1+tan²(x)

➡ sin(x)/cos(x) = tan(x)

➡ Zero product rule says that if a and b are two expressions or numbers and if ab = 0, then either a = 0 or b = 0 or both a = 0 and b = 0.

Answer 2

Answer:

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