Question

Sin (40°+x)sin(x-50°)=1

Answer 1

Решение приложено...

Answer 2

To solve the equation sin(40°+x)sin(x-50°) = 1, we can use trigonometric identities and algebraic manipulation. Here's how you can proceed:

Let's simplify the equation step by step:

1. Start by using the identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B):

   sin(40° + x)sin(x - 50°) = 1 (sin(40°)cos(x) + cos(40°)sin(x))(sin(x)cos(50°) - cos(x)sin(50°)) = 1 sin(40°)cos(x)sin(x)cos(50°) + cos(40°)sin(x)sin(x)cos(50°) - sin(40°)cos(x)cos(x)sin(50°) - cos(40°)sin(x)cos(x)sin(50°) = 1

2. Apply the double-angle identity sin(2θ) = 2sin(θ)cos(θ):

   sin(40°)cos(x)sin(x)cos(50°) + cos(40°)sin(x)sin(x)cos(50°) - sin(40°)cos(x)cos(x)sin(50°) - cos(40°)sin(x)cos(x)sin(50°) = 1 2sin(40°)sin(x)cos(x)cos(50°) + 2cos(40°)sin(x)cos(x)sin(50°) - sin(40°)cos(x)cos(x)sin(50°) - cos(40°)sin(x)cos(x)sin(50°) = 1

3. Combine like terms:

   (2sin(40°)cos(50°) - sin(40°)cos(x)cos(x) - cos(40°)sin(x)cos(x))sin(x) = 1

4. Simplify further:

   sin(40°)cos(50°) - sin(40°)cos(x)cos(x) - cos(40°)sin(x)cos(x) = 1/sin(x)

5. We know that sin(x) cannot be zero, as division by zero is undefined in mathematics. So, we can multiply both sides of the equation by sin(x):

   (sin(40°)cos(50°) - sin(40°)cos(x)cos(x) - cos(40°)sin(x)cos(x))sin(x) = (1/sin(x))sin(x) sin(40°)cos(50°)sin(x) - sin(40°)cos(x)cos(x)sin(x) - cos(40°)sin(x)cos(x)sin(x) = 1

6. Simplify further and combine like terms:

   sin(40°)cos(50°)sin(x) - sin(40°)cos(x)cos^2(x) - cos(40°)sin^2(x)cos(x) = 1

7. Rearrange the terms and use the identity sin^2(x) = 1 - cos^2(x):

   sin(40°)cos(50°)sin(x) - sin(40°)cos(x)(1 - sin^2(x)) - cos(40°)(1 - cos^2(x))cos(x) = 1 sin(40°)cos(50°)sin(x) - sin(40°)cos(x) + sin(40°)cos