Question

Sin (pi/6-x)-cos(pi/3+x) Как решить ? по какой формуле ... ? спасибо

Answer 1

Sinπ/6*cosx-cosπ/6*sinx-cosπ/3*cosx+sinπ/3*sinx=
=1/2*cosx-√3/2*sinx-1/2*cosx+√3/2*sinx=0

Answer 2

Или
sin(π/6 -x) - cos(π/3 +x) =sin( π/2 - ( π/3 +x) )  - cos(π/3 +x) =
 cos(π/3 +x) - cos(π/3 +x) =0.
------
sin( π/2 - α)  =cosα .

Answer 3

Solving the Equation sin(pi/6-x) - cos(pi/3+x)

To solve the equation sin(pi/6-x) - cos(pi/3+x), we can use trigonometric identities and properties to simplify the expression and find the solution.

Let's break down the steps to solve the equation:

Step 1: Simplify the expression using trigonometric identities. Step 2: Apply the properties of trigonometric functions to find the solution.

Let's go through each step in detail.

Step 1: Simplify the expression using trigonometric identities

We can start by using the trigonometric identities to simplify the expression.

Using the identity sin(a-b) = sin(a)cos(b) - cos(a)sin(b), we can rewrite the equation as:

sin(pi/6)cos(x) - cos(pi/6)sin(x) - cos(pi/3)cos(x) - sin(pi/3)sin(x)

Now, let's simplify further.

Using the values of sin(pi/6) = 1/2, cos(pi/6) = sqrt(3)/2, sin(pi/3) = sqrt(3)/2, and cos(pi/3) = 1/2, we can substitute these values into the equation:

(1/2)cos(x) - (sqrt(3)/2)sin(x) - (1/2)cos(x) - (sqrt(3)/2)sin(x)

Simplifying this further, we get:

-cos(x)sin(x) - sqrt(3)/2(sin(x) + cos(x))

Step 2: Apply the properties of trigonometric functions to find the solution

To find the solution, we can set the expression equal to zero and solve for x:

-cos(x)sin(x) - sqrt(3)/2(sin(x) + cos(x)) = 0

Now, let's solve for x.

We can factor out sin(x) from the expression:

sin(x)(-cos(x) - sqrt(3)/2) - sqrt(3)/2(cos(x)) = 0

Next, we can factor out cos(x) from the expression:

(-cos(x) - sqrt(3)/2)(sin(x) + sqrt(3)/2) = 0

To find the values of x that satisfy this equation, we can set each factor equal to zero and solve for x:

1. Setting -cos(x) - sqrt(3)/2 = 0: - cos(x) = -sqrt(3)/2 - Taking the inverse cosine of both sides, we get: - x = arccos(-sqrt(3)/2)

2. Setting sin(x) + sqrt(3)/2 = 0: - sin(x) = -sqrt(3)/2 - Taking the inverse sine of both sides, we get: - x = arcsin(-sqrt(3)/2)

Therefore, the solutions to the equation sin(pi/6-x) - cos(pi/3+x) are:

- x = arccos(-sqrt(3)/2) - x = arcsin(-sqrt(3)/2)

Please note that these solutions are approximate values and can be expressed in radians or degrees, depending on the context.

I hope this helps! Let me know if you have any further questions.