Find all the solutions of sin x + cos x = 1 in the interval [0, 2pie).

X = 0,  π/2 in the interval  [0, 2pi).

Step-by-step explanation:

Use the auxiliary angle method:

R sin(x + a)  = Rsin x cos a + Rcos x sin a = 1

                        sin x         +    cos x             = 1

Comparing coefficients:

R cos a = 1 and R sin a =  1, so

tan a = R sin a / R cos a  = 1

So a = π/4 radians.

Also  R^2(sin^2 a + cos^2 a) =  1^2 + 1^2 = 2

Therefore R = √2.

So √2 sin (x +π/4 = 1

sin x + π/4 = 1/√2

x  + π/4 = π/4

x = 0 radians

Also

x = 0 +  π/2 = π/2.

The first octant inside the cylinder  x2  +  y2  =  16  and below the plane z = 3. 2 sketch the solid whose volume is given by the integral. ∫ π/2. 0. ∫ 2. 0. ∫ 9−r2. 0 r dz dr dθ, and  evaluate  the integral. 3  use spherical coordinates  to  evaluate. ∫∫∫.  e  z  dv  , where  e lies between  the  spheres x2+y2+z2  = 1 and  x2  +y2  +  z2  =  4  in