Question

Review the steps of the proof of the identity

At which step was an error made?
sin (A-3) - COSA.
= A.
step 1
step 2
3п
sin
step 3
step 4
Step 1: = sinAcos
3 п
2
-
COSAsin
n ()
Step 2: = (sinA)(0) + COSA( - 1)
Step 3: = (sinA)(0) + (1) (COSA)
Step 4: = 0 + COSA
Step 5: Cos A

Answer 1

step 2

and then step 3 : the error neutralized the error of step 2

Explanation:

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

and because

sin(-b) = -sin(b)

cos(-b) = cos(b)

we have

sin(a - b) = sin(a)cos(-b) + cos(a)sin(-b) =

= sin(a)cos(b) - cos(a)sin(b)

3pi/2 = 270° or -90°.

sin(3pi/2) = sin(270) = sin(-90) = -1

that means, it is the full radius straight down from the center of the circle.

cos(3pi/2) = cos(270) = cos(-90) = cos(90) = 0

so, step 1 is correct :

sin(A - 3pi/2) = sin(A)cos(3pi/2) - cos(A)sin(3pi/2) =

= sin(A)×0 - cos(A)×(-1) (correct step 2)

but as we see, the provided step 2 is incorrect.

it should have been the indicated

sin(A)×0 - cos(A)×(-1)

and NOT the provided

sin(A)×0 + cos(A)×(-1)

step 3 based on the erroneous step 2 should then have been

sin(A)×0 - (1)cos(A)

but instead another error with the sign was made that neutralized the error of step 2 and we got after this second mistake by pure chance the overall correct step 3

sin(A)×0 + (1)cos(A)

so, again, the first error was made in step 2.

but technically, there was also a consecutive error made in step 3 to bring everything back to the correct approach.