Question

sinx-5cosx=7

Answer 1

Решение задания смотри на фотографии

Answer 2

To solve the equation sin(x) - 5cos(x) = 7, we can use trigonometric identities to simplify it. First, we can rewrite the cosine term in terms of sine using the Pythagorean identity:

cos(x) = sqrt(1 - sin^2(x))

Now, the equation becomes:

sin(x) - 5 * sqrt(1 - sin^2(x)) = 7

Next, let's isolate sin(x) on one side of the equation:

sin(x) - 5 * sqrt(1 - sin^2(x)) = 7

sin(x) = 7 + 5 * sqrt(1 - sin^2(x))

Now, we can square both sides of the equation to eliminate the square root:

sin^2(x) = (7 + 5 * sqrt(1 - sin^2(x)))^2

Let's solve for sin^2(x) first:

sin^2(x) = (7 + 5 * sqrt(1 - sin^2(x)))^2

sin^2(x) = 49 + 70 * sqrt(1 - sin^2(x)) + 25 * (1 - sin^2(x))

Now, we can isolate the sin^2(x) terms:

sin^2(x) - 25 * sin^2(x) = 49 + 70 * sqrt(1 - sin^2(x)) - 25

Combine like terms:

-24 * sin^2(x) = 24 + 70 * sqrt(1 - sin^2(x))

Now, divide both sides by -24:

sin^2(x) = (24 + 70 * sqrt(1 - sin^2(x))) / -24

sin^2(x) = - (24 + 70 * sqrt(1 - sin^2(x))) / 24

sin^2(x) = - (1 + (35/12) * sqrt(1 - sin^2(x)))

Now, square both sides of the equation again:

sin^4(x) = [-(1 + (35/12) * sqrt(1 - sin^2(x)))]^2

sin^4(x) = (1 + (35/12) * sqrt(1 - sin^2(x)))^2

Now, let's solve for sin^4(x):

sin^4(x) = 1 + 2 * (35/12) * sqrt(1 - sin^2(x)) + (35/12)^2 * (1 - sin^2(x))

sin^4(x) = 1 + (35/6) * sqrt(1 - sin^2(x)) + (1225/144) * (1 - sin^2(x))

sin^4(x) = 1 + (35/6) * sqrt(1 - sin^2(x)) + (1225/144) - (1225/144) * sin^2(x)

Now, this is a quartic equation in sin^2(x). You can solve it numerically to find the values of sin^2(x) and then take the square root to find sin(x). It may require the use of a calculator or computer software to solve for the values of sin(x).