Question

Sin 2 x - cos x = 0​

Answer 1

Ответ:

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Answer 2

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Answer 3

To solve the equation sin(2x) - cos(x) = 0, we can use trigonometric identities and algebraic manipulation. Let's break it down step by step:

1. Start with the equation: sin(2x) - cos(x) = 0.

2. We'll use the double-angle identity for sine: sin(2x) = 2sin(x)cos(x). Substituting this into the equation, we get: 2sin(x)cos(x) - cos(x) = 0.

3. Factor out cos(x) from the terms: cos(x)(2sin(x) - 1) = 0.

4. Now, we have two possibilities: a) cos(x) = 0. In this case, x can take values where cos(x) equals zero, such as x = π/2, 3π/2, 5π/2, etc. b) 2sin(x) - 1 = 0. Solve this equation separately.

5. Solve 2sin(x) - 1 = 0: Add 1 to both sides: 2sin(x) = 1. Divide both sides by 2: sin(x) = 1/2.

6. Now, we need to find the values of x where sin(x) equals 1/2. These values are typically associated with angles on the unit circle. The solutions are x = π/6 + 2πn and x = 5π/6 + 2πn, where n is an integer.

Combining both sets of solutions, the complete set of solutions for the equation sin(2x) - cos(x) = 0 is: x = π/2, 3π/2, 5π/2, π/6 + 2πn, 5π/6 + 2πn (where n is an integer).