4+5 tg^2x cos^2x , sin x=0.4

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вопрос

To evaluate the expression 4 + 5tan^2(x)cos^2(x) when sin(x) = 0.4, first we need to find the values of tan(x) and cos(x) using the given information.

Given that sin(x) = 0.4, we know that sin(x) = opposite/hypotenuse. Let's assume the hypotenuse is 5 (as it is easier to work with), so the opposite side would be 0.4*5 = 2.

Now, using this information, we can calculate the adjacent side using the Pythagorean theorem:

hypotenuse^2 = opposite^2 + adjacent^2
5^2 = 2^2 + adjacent^2
25 = 4 + adjacent^2
adjacent^2 = 21
adjacent = sqrt(21) ≈ 4.58

Now, we can find the values of tan(x) and cos(x):

tan(x) = opposite/adjacent = 2/4.58 ≈ 0.4378
cos(x) = adjacent/hypotenuse = 4.58/5 ≈ 0.9160

Finally, substitute these values into the expression:

4 + 5(tan^2(x))(cos^2(x))
4 + 5(0.4378^2)(0.9160^2)
4 + 5(0.1915)(0.8389)
4 + 0.8048
= 4.8048

Therefore, the value of 4 + 5tan^2(x)cos^2(x) when sin(x) = 0.4 is approximately 4.8048.