Question

Sqrt(2)sinx+sqrt(2)cos=2

Answer 1

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Answer 2

To solve the equation sqrt(2)sin(x) + sqrt(2)cos(x) = 2, we can first simplify it by dividing both sides by sqrt(2):

sin(x) + cos(x) = 2/sqrt(2)

Now, simplify the right-hand side:

2/sqrt(2) = 2 * sqrt(2) / 2 = sqrt(2)

So, the equation becomes:

sin(x) + cos(x) = sqrt(2)

Now, to solve this equation, we can use trigonometric identities. One useful identity is:

sin(π/4) = cos(π/4) = 1/√2

Now, we can rewrite sqrt(2) as sin(π/4) * √2:

sqrt(2) = sin(π/4) * √2

Thus, the equation becomes:

sin(x) + cos(x) = sin(π/4) * √2

We can now compare the left-hand side with the right-hand side:

sin(x) = sin(π/4) * √2

Now, we need to find all the values of x that satisfy this equation. Since sin(x) is a periodic function with a period of 2π, the general solution to this equation is:

x = π/4 + 2nπ (where n is an integer)

This means that any value of x that is π/4 plus an integer multiple of 2π will satisfy the original equation. For example, some of the solutions for x are:

x = π/4, 9π/4, 17π/4, -7π/4, -15π/4, ...

Note that the trigonometric functions are periodic, and there are infinitely many solutions to this equation due to their periodic nature.