How do you find the general solutions for $sin theta - cos theta = 1$ ?

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Answer 1 · 2015-08-17T03:49:17

Solve sin x - cos x = 1

Ans: $pi and pi/2$

Apply the trig identity: $sin x - cos x = sqrt2.sin (x - pi/4)$

$sqrt2sin (x - pi/4) = 1$
$sin (x - pi/4) = 1/sqrt2 = sqrt2/2$
Trig table gives -->
$(x - pi/4) = pi/4$ and $(x - pi/4) = (3pi)/4$

a. $x - pi/4 = pi/4$ --> $x = (2pi)/ 4 = pi/2$

b. $x - pi/4 = 3pi/4$ --> $x = (3pi)/4 + pi/ 4 = pi$
Check
$x = pi/2$ --> $sin (pi/2) - cos (pi/2) = 1$ OK

$x = pi$ --> $sin pi - cos pi = 1$ --> 0 - (-1) = 1 OK

How do you find the general solutions for sin theta - cos theta = 1sin theta - cos theta = 1?