Question

Sin (x/3 + п/6) = √ 2/2

Answer 1

Ответ:

Sin (x/3 + п/6) = √ 2/2

Sin (x/3 + п/6) = √ 1

Sin (x/3 + п/6) = 1

х/3+п/6=п/2+2kп, k € z

2x+п=3п+12kп, k € z

2x=2п+12kп,k€a

x=п+6kп, k€z

Альтернативная форма:

х=180°+k×1080°, k€Z

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Answer 2

To solve the equation sin(x/3 + π/6) = √2/2, we can use the properties of the sine function and trigonometric identities. Let's break down the steps:

1. Rewrite √2/2 as sin(π/4): The value √2/2 is equal to sin(π/4). This is because sin(π/4) = √2/2 is one of the standard values for the sine function.

2. Rewrite the given equation using the sine addition formula: The sine addition formula states that sin(a + b) = sin(a)cos(b) + cos(a)sin(b). Applying this formula to our equation, we have: sin(x/3)cos(π/6) + cos(x/3)sin(π/6) = sin(π/4).

3. Simplify the equation: sin(x/3)(√3/2) + cos(x/3)(1/2) = √2/2.

4. Multiply through by 2 to clear the denominators: 2sin(x/3)(√3/2) + 2cos(x/3)(1/2) = 2(√2/2).

   Simplifying further: √3sin(x/3) + cos(x/3) = √2.

5. Use the Pythagorean identity sin²θ + cos²θ = 1: Rearrange the equation to isolate sin(x/3): sin(x/3) = √2 - √3cos(x/3).

   Square both sides: sin²(x/3) = (2 - 2√3cos(x/3) + 3cos²(x/3)).

   Apply the Pythagorean identity: 1 - cos²(x/3) = 2 - 2√3cos(x/3) + 3cos²(x/3).

   Simplify and rearrange the equation: 4cos²(x/3) - 2√3cos(x/3) + 1 = 0.

6. Solve the quadratic equation for cos(x/3): Let's substitute z = cos(x/3): 4z² - 2√3z + 1 = 0.

   Using the quadratic formula: z = (-(-2√3) ± √((-2√3)² - 4(4)(1))) / (2(4)). z = (√3 ± √(12 - 16)) / 8. z = (√3 ± √(-4)) / 8.

   Since the square root of a negative number is not a real number, there are no real solutions for cos(x/3).

Therefore, the equation sin(x/3 + π/6) = √2/2 has no real solutions.