Question

√3 sinx-cosx=1 Решите, пожалуйста, уравнение. Преобразование тригонометрических выражений, 10 класс.

Answer 1

Разделяем на два
√3/2sinx-1/2cosx=1/2
sinx*cos(π/6)-sin(π/6)cosx=1/2
sin(x-π/6)=1/2
x-π/6=(-1)^k*π/6+πk,k∈Z
x=(-1)^k*π/3+πk,k∈Z

Answer 2

Solving the Trigonometric Equation

To solve the equation √3 + sinx - cosx = 1, we can use trigonometric identities and algebraic manipulation to simplify and solve for the value of x.

First, let's start by manipulating the given equation using trigonometric identities and algebraic operations.

We can rewrite the equation as: sinx + cosx = 1 - √3.

Now, let's use the identity sin^2x + cos^2x = 1 to simplify the equation further.

We know that sin^2x + cos^2x = 1, so sinx = √(1 - cos^2x).

Substituting this into the equation, we get: √(1 - cos^2x) + cosx = 1 - √3.

Let's denote cosx as t for simplicity. Then the equation becomes: √(1 - t^2) + t = 1 - √3.

We can now solve for t and then find the corresponding values of x.

Solution Steps

1. Square both sides of the equation to eliminate the square root. 2. Rearrange the terms to isolate t. 3. Solve for t. 4. Find the corresponding values of x.

Solution

1. Square both sides of the equation: (√(1 - t^2))^2 + 2√(1 - t^2)t + t^2 = (1 - √3)^2.

2. Rearrange the terms to isolate t: 1 - t^2 + 2√(1 - t^2)t + t^2 = 1 - 2√3 + 3.

3. Solve for t: 2√(1 - t^2)t = -2√3.

4. Find the corresponding values of x: After finding the value of t, we can use the relation t = cosx to find the corresponding values of x.

This approach will help us find the solutions for the given trigonometric equation.