I have an equation where solutions to cos(x)=sin(x) are needed. I have been told that sin(x)/cos(x) is equal to tan(x) but also that sin(x)/cos(x)=1. For which value will sin(x)/cos(x) result in 1?
Another way to do it: On the unit circle, cos x corresponds to the x-coordinate and sin x corresponds to the y-coordinate. Therefore, $\displaystyle \sin x = \cos x$ if and only if the point on the unit circle is such that the x-coordinate equals the y-coordinate, or the point lies on the line y=x. Where does the unit circle intersect the line y=x?
Of course, $\displaystyle \frac{sin(x)}{cos(x)}= 1$ is the same as $\displaystyle sin(x)= cos(x)$. Since, in basic terms, "sine" and "cosine" refer to opposite angles in a right triangle, if they are equal those angles must be the same- and since those those angles must add to 90 degrees, it should be easy to see what the angle must be. In more advanced terms, where the angles are not in a right triangle, we can use the "circle" formulation and see that adding 180 degrees will also be an angle.