Question

4 sin(x) - 6 cos(x) = 1

Answer 1

$4 sinx - 6 cosx = 1$

$2 sinx - 3 cosx = 0.5$

Решим данное уравнение с помощью метода введения вспомогательного угла. 

Разделим обе части уравнения на  $\sqrt{2^2+3^2} = \sqrt{13} .$
Имеем:

$\frac{2}{ \sqrt{13} } sinx- \frac{3}{ \sqrt{13} } cosx= \frac{1}{2 \sqrt{13} }$

так как  $( \frac{2}{ \sqrt{13} } )^2+( \frac{3}{ \sqrt{13} } )^2=1,$  то примем  $\frac{2}{ \sqrt{13} }$ за косинус некоторого угла φ, а   $\frac{3}{ \sqrt{13} } -$ за синус этого же  угла.

Следовательно, уравнение примет вид:

$cos$ φ  $*sinx-sin$ φ  $*cosx= \frac{1}{2 \sqrt{13} }$

$sin( x-$ φ  $)= \frac{1}{2 \sqrt{13} }$

$x=(-1)^karcsin \frac{1}{2 \sqrt{13} } +$ φ  $+ \pi k,$  $k$ ∈  $Z,$  где φ  $=arcsin \frac{3}{ \sqrt{13} }$

$x=(-1)^karcsin \frac{1}{2 \sqrt{13} } +arcsin \frac{3}{ \sqrt{13} } + \pi k,$  $k$ ∈  $Z$


  

Answer 2

To solve the trigonometric equation 4 sin(x) - 6 cos(x) = 1, we can use trigonometric identities and techniques. The general strategy is to rewrite the equation in terms of a single trigonometric function, either sine or cosine, and then solve for that function.

Let's solve the equation step by step:

Step 1: Recognize the Pythagorean Identity The Pythagorean identity is: sin^2(x) + cos^2(x) = 1

Step 2: Divide the entire equation by sqrt(4^2 + (-6)^2) Dividing the equation by sqrt(4^2 + (-6)^2) = sqrt(16 + 36) = sqrt(52) = 2*sqrt(13), we get:

(4/2sqrt(13)) * sin(x) - (6/2sqrt(13)) * cos(x) = 1/(2*sqrt(13))

Step 3: Express sin(x) and cos(x) in terms of a single trigonometric function

Divide both coefficients by 2*sqrt(13):

(sin(x))/sqrt(13) - (3cos(x))/sqrt(13) = 1/(2sqrt(13))

Now, we need to find a value a such that:

asin(x) - acos(x) = 1/(2*sqrt(13))

Let's find the value of 'a':

a = sqrt(13) So, our equation becomes:

sqrt(13)*sin(x) - sqrt(13)cos(x) = 1/(2sqrt(13))

Step 4: Use the Difference of Angles Identity The difference of angles identity for sine and cosine is: sin(A - B) = sin(A)*cos(B) - cos(A)*sin(B)

We can rewrite our equation in terms of the difference of angles identity:

sqrt(13) * [sin(x) - cos(x)] = 1 / (2 * sqrt(13))

Step 5: Isolate sin(x) - cos(x) Divide both sides by sqrt(13):

sin(x) - cos(x) = 1 / (2 * 13)

Step 6: Solve for sin(x) - cos(x) Adding cos(x) to both sides:

sin(x) = 1 / (2 * sqrt(13)) + cos(x)

Step 7: Use the Pythagorean Identity to rewrite cos(x) in terms of sin(x) The Pythagorean identity is: sin^2(x) + cos^2(x) = 1

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

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

Step 8: Substitute the value of cos(x) in terms of sin(x) sin(x) = 1 / (2 * sqrt(13)) + sqrt(1 - sin^2(x))

Step 9: Rearrange the equation to solve for sin(x) sin^2(x) = [1 / (2 * sqrt(13))]^2 + 2 * [1 / (2 * sqrt(13))] * sqrt(1 - sin^2(x)) + [sqrt(1 - sin^2(x))]^2

sin^2(x) = 1 / (52) + 1 / sqrt(13) * sqrt(1 - sin^2(x)) + 1 - sin^2(x)

sin^2(x) + sin^2(x) - 1 / sqrt(13) * sqrt(1 - sin^2(x)) = 1 / 52

Step 10: Let u = sin(x) Now, we have a quadratic equation in terms of u:

u^2 + u - 1 / sqrt(13) * sqrt(1 - u^2) - 1 / 52 = 0

This equation can be solved for u (sin(x)) using numerical methods or factoring methods, but the exact solution involves solving a high-degree polynomial and is quite complex.

As such, it might be more appropriate to use numerical methods like numerical approximation or graphing to find the approximate values of sin(x) that satisfy the original equation. If you have a specific range or interval in mind for x, I can help you find numerical approximations for sin(x) in that range.