Solve for x sin(x)+cos(x)=0

Problem

sin(x)+cos(x)=0

Solution

1. Subtract cos(x) from both sides of the equation to isolate the trigonometric terms.

sin(x)=−cos(x)

2. Divide both sides by cos(x) assuming cos(x)≠0 to express the equation in terms of the tangent function.

sin(x)/cos(x)=−1

3. Substitute the identity tan(x)=sin(x)/cos(x) into the equation.

tan(x)=−1

4. Identify the reference angle where tan(x)=1 which is π/4 Since the value is negative, x must be in the second or fourth quadrants.

x=(3*π)/4,(7*π)/4,…

5. Generalize the solution by adding multiples of the period of the tangent function, which is π

x=(3*π)/4+n*π

Final Answer

sin(x)+cos(x)=0⇒x=(3*π)/4+n*π,n∈ℤ

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