With reference to the right triangle,, explain why the expressions y = sin x and y = cos ((pi/2) - (x)) give the same results for all the values of x. Discuss whether or not this same relationship exists for angles greater than (pi/2).

sketch a right-angled triangle in quadrant I ,

labeling the two acute angles A and B
then A+B = 90° or π/2
and B = (90°-A) or (π/2 - A)

label the horizontal x and the vertical side y
sinA = y/r
cosB = y/r
so sinA = cosB = cos(π/2 - A)
or in the terminology of your question ...
sinx = cos(π/2 - x)

this relationship is called the complementary trig relationship and is true for any trig ratio with its co-trig ratio
i.e. sine ---- cosine
secant --- cosecant
tangent ---- cotangent

Making sketches in quadrants II , III, and IV, it can be shown that it is true for any size of x

e.g. is sin 150° = cos(90° - 150° ??

sin 150 = sin30 = 1/2
cos(90-150) = cos(-60) = cos60 = 1/2

how about x = 300?
sin 300 = -√3/2
cos(90-300)
= cos( - 210)
= cos 210
= -cos30
= -√3/2

ur amazing !

To understand why the expressions y = sin x and y = cos ((π/2) - x) give the same results for all values of x in a right triangle, let's break it down.

1. Definition of Sine (sin) and Cosine (cos):
In a right triangle, sin(x) is defined as the ratio between the length of the side opposite angle x and the length of the hypotenuse, while cos(x) is defined as the ratio between the length of the side adjacent to angle x and the length of the hypotenuse.

2. Relationship in a Right Triangle:
In a right triangle, one acute angle (x) is always complementary to the other acute angle (π/2 - x). Complementary angles add up to 90 degrees (or π/2 radians). This means that if one angle is x, the other angle is (π/2 - x).

3. Relationship between Sine and Cosine:
With the above complementary relationship, we can see that the side opposite angle x is equal to the side adjacent to angle (π/2 - x), and vice versa. Therefore, when we calculate sin(x) (using the side opposite angle x) and cos((π/2 - x)) (using the side adjacent to angle (π/2 - x)), we are actually determining the ratio of the same pair of sides in the right triangle.

4. Solving for sin x and cos((π/2 - x)):
Since the ratio of the same pair of sides remains constant, the values of sin x and cos((π/2 - x)) will be equal for all values of x.

Regarding angles greater than π/2, the same relationship between sin x and cos((π/2 - x)) doesn't exist. Once we move beyond π/2, the complementary relationship no longer holds, and the values of sin x and cos((π/2 - x)) will differ. In other words, sin x will represent the ratio between the side opposite the angle x and the hypotenuse, while cos((π/2 - x)) will represent the ratio between the side adjacent to (π/2 - x) and the hypotenuse.