To solve the equation \(\cos(t) - \sin(t) = 1\) for \(t\) in the interval \([0, 2\pi)\), we can follow these steps:
1. Rewrite the equation:
[tex]\[
\cos(t) - \sin(t) = 1
\][/tex]
2. Square both sides to eliminate the trigonometric functions:
[tex]\[
(\cos(t) - \sin(t))^2 = 1^2
\][/tex]
3. Expand the left-hand side:
[tex]\[
\cos^2(t) - 2\cos(t)\sin(t) + \sin^2(t) = 1
\][/tex]
4. Use the Pythagorean identity, \( \cos^2(t) + \sin^2(t) = 1 \):
[tex]\[
1 - 2\cos(t)\sin(t) = 1
\][/tex]
5. Simplify the equation:
[tex]\[
-2\cos(t)\sin(t) = 0
\][/tex]
6. Factor the expression:
[tex]\[
-2\cos(t)\sin(t) = 0
\][/tex]
[tex]\[
\cos(t)\sin(t) = 0
\][/tex]
7. Set each factor equal to zero and solve for \( t \):
[tex]\[
\cos(t) = 0 \quad \text{or} \quad \sin(t) = 0
\][/tex]
8. Solve \( \cos(t) = 0 \):
[tex]\[
t = \frac{\pi}{2}, \frac{3\pi}{2} \quad (\text{in the interval } [0, 2\pi))
\][/tex]
9. Solve \( \sin(t) = 0 \):
[tex]\[
t = 0, \pi, 2\pi \quad (\text{in the interval } [0, 2\pi))
\][/tex]
10. Check each solution to see if it satisfies the original equation \( \cos(t) - \sin(t) = 1 \):
- For \( t = 0 \):
[tex]\[
\cos(0) - \sin(0) = 1 - 0 = 1 \quad \text{(True)}
\][/tex]
- For \( t = \frac{\pi}{2} \):
[tex]\[
\cos\left(\frac{\pi}{2}\right) - \sin\left(\frac{\pi}{2}\right) = 0 - 1 = -1 \quad \text{(False)}
\][/tex]
- For \( t = \pi \):
[tex]\[
\cos(\pi) - \sin(\pi) = -1 - 0 = -1 \quad \text{(False)}
\][/tex]
- For \( t = \frac{3\pi}{2} \):
[tex]\[
\cos\left(\frac{3\pi}{2}\right) - \sin\left(\frac{3\pi}{2}\right) = 0 - (-1) = 1 \quad \text{(True)}
\][/tex]
- For \( t = 2\pi \):
[tex]\[
\cos(2\pi) - \sin(2\pi) = 1 - 0 = 1 \quad \text{(True)}
\][/tex]
11. Gather all the valid solutions:
The values of \( t \) that satisfy the equation \(\cos(t) - \sin(t) = 1\) in the interval \([0, 2\pi)\) are:
[tex]\[
t = 0, \frac{3\pi}{2}, 2\pi
\][/tex]
Therefore, the solutions to the equation [tex]\(\cos(t) - \sin(t) = 1\)[/tex] in the interval [tex]\([0, 2\pi)\)[/tex] are [tex]\( t = 0 \)[/tex], [tex]\( \frac{3\pi}{2} \)[/tex], and [tex]\( 2\pi \)[/tex].
