Answer :
Certainly! We're asked to find the terminal point \( P(x, y) \) on the unit circle determined by the angle \( t = -\frac{3\pi}{4} \).
### Step-by-Step Solution:
1. Understanding the Unit Circle:
- The unit circle is a circle with a radius of 1 centered at the origin \((0,0)\) on a coordinate plane. Each point on the unit circle has coordinates \((x, y)\) that satisfy the equation \( x^2 + y^2 = 1 \).
2. Angle Representation:
- The angle \( t = -\frac{3\pi}{4} \) is measured in radians. A negative angle indicates that we measure the angle clockwise starting from the positive x-axis.
3. Locating the Angle:
- We start from the positive x-axis and rotate clockwise by \( \frac{3\pi}{4} \) radians.
- \( \frac{3\pi}{4} \) radians is equivalent to 135°. Since we are rotating clockwise, we are effectively locating the angle at \( 225° \) counter-clockwise from the positive x-axis (as subtracting 135° is equivalent to adding 225° in the positive direction).
4. Using Trigonometric Functions:
- For an angle \( \theta \) on the unit circle, the coordinates \((x, y)\) are given by \( x = \cos(\theta) \) and \( y = \sin(\theta) \).
- Here, \( \theta = -\frac{3\pi}{4} \). Plugging this angle into the trigonometric functions:
[tex]\[ x = \cos\left(-\frac{3\pi}{4}\right) \quad \text{and} \quad y = \sin\left(-\frac{3\pi}{4}\right) \][/tex]
5. Calculating \( \cos \) and \( \sin \) Values:
- The cosine and sine of \( -\frac{3\pi}{4} \) radians (or \( 225° \)) can be determined:
- \( \cos\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)
- \( \sin\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)
6. Final Coordinates:
- Thus, the coordinates for the terminal point \( P(x, y) \) on the unit circle are:
[tex]\[ x = -0.7071067811865475 \quad \text{and} \quad y = -0.7071067811865476 \][/tex]
### Conclusion:
The terminal point \( P(x, y) \) on the unit circle determined by the angle \( t = -\frac{3\pi}{4} \) is:
[tex]\[ P\left( -0.7071067811865475, -0.7071067811865476 \right) \][/tex]
This detailed explanation helps us understand the steps involved in finding the terminal point on the unit circle for a given angle.
### Step-by-Step Solution:
1. Understanding the Unit Circle:
- The unit circle is a circle with a radius of 1 centered at the origin \((0,0)\) on a coordinate plane. Each point on the unit circle has coordinates \((x, y)\) that satisfy the equation \( x^2 + y^2 = 1 \).
2. Angle Representation:
- The angle \( t = -\frac{3\pi}{4} \) is measured in radians. A negative angle indicates that we measure the angle clockwise starting from the positive x-axis.
3. Locating the Angle:
- We start from the positive x-axis and rotate clockwise by \( \frac{3\pi}{4} \) radians.
- \( \frac{3\pi}{4} \) radians is equivalent to 135°. Since we are rotating clockwise, we are effectively locating the angle at \( 225° \) counter-clockwise from the positive x-axis (as subtracting 135° is equivalent to adding 225° in the positive direction).
4. Using Trigonometric Functions:
- For an angle \( \theta \) on the unit circle, the coordinates \((x, y)\) are given by \( x = \cos(\theta) \) and \( y = \sin(\theta) \).
- Here, \( \theta = -\frac{3\pi}{4} \). Plugging this angle into the trigonometric functions:
[tex]\[ x = \cos\left(-\frac{3\pi}{4}\right) \quad \text{and} \quad y = \sin\left(-\frac{3\pi}{4}\right) \][/tex]
5. Calculating \( \cos \) and \( \sin \) Values:
- The cosine and sine of \( -\frac{3\pi}{4} \) radians (or \( 225° \)) can be determined:
- \( \cos\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)
- \( \sin\left(-\frac{3\pi}{4}\right) = -\frac{\sqrt{2}}{2} \)
6. Final Coordinates:
- Thus, the coordinates for the terminal point \( P(x, y) \) on the unit circle are:
[tex]\[ x = -0.7071067811865475 \quad \text{and} \quad y = -0.7071067811865476 \][/tex]
### Conclusion:
The terminal point \( P(x, y) \) on the unit circle determined by the angle \( t = -\frac{3\pi}{4} \) is:
[tex]\[ P\left( -0.7071067811865475, -0.7071067811865476 \right) \][/tex]
This detailed explanation helps us understand the steps involved in finding the terminal point on the unit circle for a given angle.
