how are the following relations true? S(theta + pi/2) = C(theta) and C(theta + pi/2) = -S(theta)?
To understand why these relations are true, let's start by revisiting the definitions of sine (S) and cosine (C) functions:
- Sine (S): In a right triangle, the sine of an angle θ is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse.
- Cosine (C): In a right triangle, the cosine of an angle θ is defined as the ratio of the length of the adjacent side to the length of the hypotenuse.
Now, let's consider a unit circle, which is defined as a circle with a radius of 1 unit, centered at the origin (0, 0) on a Cartesian coordinate system. The angles on a unit circle are measured in radians.
1. S(theta + pi/2) = C(theta):
When we add pi/2 (90 degrees) to an angle θ and take the sine of the result, it is equivalent to taking the cosine of the original angle θ.
To understand why, imagine the unit circle with an angle θ. The point on the unit circle corresponding to this angle can be represented as (C(theta), S(theta)).
Now, if we add pi/2 to θ, the new angle becomes (θ + pi/2), and the corresponding point on the unit circle changes to (C(theta + pi/2), S(theta + pi/2)).
Since we are dealing with a unit circle, the length of the radius is 1. The x-coordinate of a point on the unit circle represents the cosine value, and the y-coordinate represents the sine value. Therefore, (C(theta), S(theta)) and (S(theta + pi/2), C(theta + pi/2)) are the same points on the unit circle, just with their coordinates swapped.
So, S(theta + pi/2) = C(theta).
2. C(theta + pi/2) = -S(theta):
Similarly, when we add pi/2 (90 degrees) to an angle θ and take the cosine of the result, it is equivalent to taking the negative sine of the original angle θ.
Again, consider the unit circle with the angle θ. The point on the unit circle corresponding to this angle is (C(theta), S(theta)).
By adding pi/2 to θ, the new angle becomes (θ + pi/2), and the corresponding point on the unit circle changes to (C(theta + pi/2), S(theta + pi/2)).
Since the x-coordinate represents the cosine and the y-coordinate represents the sine, the two points (C(theta), S(theta)) and (C(theta + pi/2), S(theta + pi/2)) are the same points on the unit circle, but the y-coordinate of the latter is the negative value of the x-coordinate of the former.
Therefore, C(theta + pi/2) = -S(theta).
In summary, the relations S(theta + pi/2) = C(theta) and C(theta + pi/2) = -S(theta) hold because they derive from the geometric properties of the unit circle and the definitions of sine and cosine functions.