Question

6 sin альфа -4 cos альфа /4 sin альфа -10 cos альфа =-1

Answer 1

$\frac{6Sin\alpha-4Cos\alpha}{4Sin\alpha-10Cos\alpha}=-1\\\\6Sin\alpha-4Cos\alpha=-4Sin\alpha+10Cos\alpha\\\\6Sin\alpha-4Cos\alpha+4Sin\alpha-10Cos\alpha=0\\\\10Sin\alpha -14Cos\alpha=0\\\\5Sin\alpha -7Cos\alpha=0|:Cos\alpha\neq0\\\\5tg\alpha-7=0\\\\tg\alpha=\frac{7}{5} \\\\\alpha=arctg\frac{7}{5}+\pi n,n \in Z\\\\\alpha=arctg1,4+\pi n,n\in Z$

Answer 2

To solve the equation:

(6sin(α) - 4cos(α)) / (4sin(α) - 10cos(α)) = -1

We can simplify it by multiplying both sides of the equation by the denominator (4sin(α) - 10cos(α)) to eliminate the fraction:

(6sin(α) - 4cos(α)) = -1 * (4sin(α) - 10cos(α))

Expanding the right side:

6sin(α) - 4cos(α) = -4sin(α) + 10cos(α)

Now, let's group the terms with sin(α) and cos(α) respectively:

(6sin(α) + 4sin(α)) = (10cos(α) + 4cos(α))

10sin(α) = 14cos(α)

Next, we can divide both sides of the equation by 2 to simplify further:

5sin(α) = 7cos(α)

Now, we can use the identity tan(α) = sin(α) / cos(α) to rewrite the equation:

5tan(α) = 7

Dividing both sides by 5:

tan(α) = 7/5

Finally, we can find the value of α by taking the arctan of both sides:

α = arctan(7/5)

Using a calculator or reference table, we find that α ≈ 54.46 degrees or approximately 0.952 radian (rounded to three decimal places).