Question

SQRT(3)sin(x+pi/4) + sin(x+pi/4)=0

Answer 1

Уравнение имеет решение только тогда, когда
sin(x+pi/4)=0
x+pi/4=pi*k, k∈Z
x=-pi/4+pi*k, k∈Z

Answer 2

To solve the trigonometric equation SQRT(3)sin(x+pi/4) + sin(x+pi/4) = 0, we can use some trigonometric identities to simplify it and find the solutions. Let's go step by step:

Step 1: Combine the two terms on the left side since they have a common factor of sin(x+pi/4):

SQRT(3)sin(x+pi/4) + sin(x+pi/4) = sin(x+pi/4) * (SQRT(3) + 1) = 0

Step 2: Set each factor equal to zero:

1. sin(x+pi/4) = 0
2. SQRT(3) + 1 = 0

Step 3: Solve each equation separately:

1. sin(x+pi/4) = 0

The sine function is zero at specific angles. The general form of the solution for sin(A) = 0 is A = n*pi, where n is an integer. So in our case:

x + pi/4 = n*pi

To isolate x, subtract pi/4 from both sides:

x = n*pi - pi/4

2. SQRT(3) + 1 = 0

This equation has no real solutions since the sum of the square root of 3 and 1 is positive and cannot be equal to zero.

So, the final solution to the trigonometric equation is:

x = n*pi - pi/4, where n is an integer.