Question

How do you solve `1 + sin(x) = cos(x)`?

Answer 1

Answers: `pi, (3pi)/2`

Explanation:

Use the trig formula:
`sin a - cos a = sqrt2sin (a + pi/4)`
`sin x - cos x = -1`
`sqrt2sin (x + pi/4) = - 1`
`sin (x + pi/4) = - 1/sqrt2 = -sqrt2/2`
Trig table and unit circle -->there are 2 solutions:
a. `x + pi/4 = -pi/4`
`x = - pi/4 - pi/4 = -pi/2`, or `3pi/2` (co-terminal)
b. `x + pi/4 = pi - (-pi/4) = (5pi)/4`
x = `(5pi)/4 - pi/4 = pi`
Answers for `(0, 2pi)`:
`pi, (3pi)/2`
Check .
`x = pi` --> sin x = 0 -> cos x = -1 --> - 1 + 0 = -1 . OK
`x = (3pi)/2` --> sin x = -1 -> cos x = 0 --> -1 + 0 = - 1. OK