Why do we use the unit circle to solve for sin and cos I know that in a unit circle where the radius is always one, sin is equal to y and cos is equal to x. But why do we use these values even when the radius or the hypothenuse of the triangle isn't equal to one. We also say that sin(π/2) = 1, but this only works if the hypothenuse of the triangle is one. What if it isn't. Can someone clear this confusion.
P.S. I have just started studying trigonometry.
 A: 
I have just started studying trigonometry.

There is a clear difference between the study of such functions as the sine and cosine, within the realm of Trigonometry or Analytical Geometry, and the study of the sine and cosine functions in Calculus (A.K.A. Real Analysis).
In Trigonometry or Analytical Geometry, the most important thing is that the definitions seem sensible.  That is why, within the realm of Trigonometry or Analytical Geometry, the domain of the sine and cosine functions are angles.  This means that you apply these functions in any arbitrary right triangle, regardless of the length of the hypotenuse.  The sine and cosine functions then measure the ratio between the legs of the right triangle versus the hypotenuse of the right triangle.  This approach makes sense.
In Calculus, these sensible definitions are altered to facilitate solving special types of problems, which you probably haven't encountered yet, and don't need to be concerned about right now.  The alterations are counter intuitive.  The domain of the sine and cosine functions are Real numbers, that as your question has indicated, are closely associated with the unit circle.
As a further alteration, you are introduced to the somewhat bizarre and somewhat ambiguous notion of radians.  For a student in transition, you will end up starting with the idea that radians measure angles, and that $(360)$ degrees equals $(2\pi)$ radians.  This a stepping stone to the definition of the sine and cosine functions in Calculus, and is based on the idea that the circumference of the unit circle is $(2\pi).$
As you complete the transition, from regarding
Trig functions in Analytical Geometry/Trigonometry, to regarding Trig functions in Calculus, the domain of the Trig functions stops being angles, and starts being dimensionless Real numbers that are associated with the arc length.
For example, $(45^\circ)$ corresponds to $(1/8)$ of a revolution, so it is equivalently regarded as $(\pi/4)$.  This means that in Calculus, the $\sin(\pi/4) = (1/\sqrt{2}).$  Note that I said $\sin(\pi/4)$, rather than $\sin(\pi/4)$ radians.
The reason that you are confused is that you are caught in the gap between Analytical Geometry/Trigonometry where there is no such thing as a radian, and angles are measured in degrees, and Calculus.
There is even further confusion.  When you first are introduced to the term radian it is natural to regard it as referring to an angle.  In Calculus, bizarrely, a radian refers to the dimensionless proportion between a certain arc length and the arc length of a complete revolution, namely $(2\pi)$.
So, when a Calculus student uses the term radians, he is not referring to an angle, but instead is referring to the dimensionless proportion between a certain arc length of a unit circle, and the circumference of the unit circle.  This means that to a Calculus student, the equations $\sin(\pi/4) = (1/\sqrt{2})$ and $\sin(\pi/4) ~\text{radians}~ = (1/\sqrt{2})$ are equivalent equations.
There is no way that you could not be confused by all of this, just as I was.  The best approach, to clarify the situation in your mind, is to explain your confusion to a college Math professor, rather than a high school Math teacher, and have the professor explain things.
A: The cosine and sine functions are defined on the unit circle. The reason for this is that when working with similar triangles you often want to figure out their relative scaling and the easiest number to multiply by is $1$. So if the length of the hypotenuse is $r$ I can say that the horizontal and vertical sides are exactly $r$ times the cosine and $r$ times the sine respectively. If it was defined on say, the circle of radius $2$ then we'd have to divide everything by $2$ and it makes the calculation needlessly complicated.
