Question

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).

Answer 1

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

Answer 2

ur amazing !