Solve for x tan(x)=- square root of 2sin(x)

Problem

tan(x)=−√(,2)*sin(x)

Solution

1. Rewrite the tangent function in terms of sine and cosine using the identity tan(x)=sin(x)/cos(x)

sin(x)/cos(x)=−√(,2)*sin(x)

2. Move all terms to one side of the equation to set it equal to zero.

sin(x)/cos(x)+√(,2)*sin(x)=0

3. Factor out the common term sin(x) from the expression.

sin(x)*(1/cos(x)+√(,2))=0

4. Apply the zero product property by setting each factor equal to zero.

sin(x)=0

1/cos(x)+√(,2)=0

5. Solve the first equation for x The sine function is zero at integer multiples of π

x=n*π

6. Isolate the cosine term in the second equation.

1/cos(x)=−√(,2)

cos(x)=−1/√(,2)

7. Rationalize the denominator to recognize the standard value.

cos(x)=−√(,2)/2

8. Determine the values of x where the cosine is −√(,2)/2 This occurs in the second and third quadrants.

x=(3*π)/4+2*n*π

x=(5*π)/4+2*n*π

Final Answer

tan(x)=−√(,2)*sin(x)⇒x=n*π,(3*π)/4+2*n*π,(5*π)/4+2*n*π

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