Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Discover comprehensive solutions to your questions from a wide network of experts on our user-friendly platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
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].
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].
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.