Trigonometric Equations

The equation

%5Csin%20(x)%3D%5Ccos%20(y)

represents a set of points

(x%2C%20y)

in the plane. We can rewrite this equation using the trigonometric identity

%5Ccos%20(y)%3D%5Csin%20%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D-y%5Cright)

, which gives us:$

%5Csin%20(x)%3D%5Csin%20%5Cleft(%5Cfrac%7B%5Cpi%7D%7B2%7D-y%5Cright)

The%20solutions%20to%20this%20equation%20are%20given%20by%20the%20general%20formula%20for%20sine%20function%20equality

x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D-y%2B2%20k%20%5Cpi%20%5Cquad%20%5Ctext%20%7B%20or%20%7D%20%5Cquad%20x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D%2By%2B(2%20k%2B1)%20%5Cpi

Where

is%20any%20integer.%20These%20equations%20describe%20a%20set%20of%20parallel%20lines%20in%20the

x y

-plane.%20The%20first%20equation%20represents%20lines%20of%20the%20form

y=\frac{\pi}{2}-x+2 k \pi

%2C%20and%20the%20second%20equation%20represents%20lines%20of%20the%20form

y=x-\frac{3 \pi}{2}-2 k \pi

.So%2C%20the%20equation

\sin (x)=\cos (y)$ represents a series of parallel lines with specific separations in the plane.

Chapter Questions
1 MCQs