Question

5/cosx-2/sinx=10/sin2x

Answer 1

=5/cosx - 2/sinx = 10/2sinxcosx

5/cosx - 2/sinx - 5/sinxcosx = 0

(5sinx-2cosx-5)/cosxsinx = 0

5sinx - 2cosx = 5

5*2t/(1+t^2) - 2* (1-t^2)/(1+t^2)=5

t1= 7/3     tg(x/2)=7/3   x=2arctg(7/3)+2pk

t2= 1     tg(x/2)=1    x=p/2 +2pk  (не удовл.)

Ответ: x=2arctg(7/3)+2pk

Answer 2

To solve the trigonometric equation 5/cos(x) - 2/sin(x) = 10/sin(2x), we can follow these steps:

Step 1: Simplify the equation using trigonometric identities. Step 2: Convert all trigonometric functions to sine and cosine. Step 3: Solve for x.

Let's go through each step:

Step 1: Simplify the equation using trigonometric identities.

We'll first simplify the right-hand side (RHS) of the equation using the double-angle identity for sine: sin(2x) = 2sin(x)cos(x)

Now, the equation becomes: 5/cos(x) - 2/sin(x) = 10/(2sin(x)cos(x))

Step 2: Convert all trigonometric functions to sine and cosine.

To convert the terms on the left-hand side (LHS) to sine and cosine, we'll use the following trigonometric identities: 1/cos(x) = sec(x) 1/sin(x) = csc(x)

Now, the equation becomes: 5sec(x) - 2csc(x) = 10/(2sin(x)cos(x))

Step 3: Solve for x.

To simplify the right-hand side (RHS) further, we'll use the double-angle identity for sine once again: 10/(2sin(x)cos(x)) = 10/(2 * 2sin(x)cos(x))

10/(2 * 2sin(x)cos(x)) = 5/(sin(x)cos(x))

Now, the equation becomes: 5sec(x) - 2csc(x) = 5/(sin(x)cos(x))

Next, we'll express sec(x) and csc(x) in terms of sine and cosine: sec(x) = 1/cos(x) csc(x) = 1/sin(x)

Now, the equation becomes: 5/(cos(x)) - 2/(sin(x)) = 5/(sin(x)cos(x))

To get rid of the fractions, let's multiply the entire equation by sin(x)cos(x): 5 * sin(x)cos(x) - 2 * sin(x)cos(x) = 5

Now, factor out sin(x)cos(x): sin(x)cos(x) * (5 - 2) = 5

Simplify: 3sin(x)cos(x) = 5

Now, we can use the double-angle identity for sine (2sin(x)cos(x) = sin(2x)) to rewrite the equation: 3 * sin(2x) = 5

Finally, solve for sin(2x): sin(2x) = 5/3

There is no real solution for sin(2x) because the range of sine function is -1 to 1, and 5/3 is outside that range. Therefore, the original equation has no solution in the real domain.