To find the exact trigonometric ratios for the angle x whose radian measure is 3π/4, we first need to determine the position of this angle on the unit circle.
1. **Locate the Angle**: The angle 3π/4 is in the second quadrant. It is equivalent to 180° – 45°, or 135° in degrees.
2. **Reference Angle**: In the second quadrant, the reference angle is found by subtracting 3π/4 from π (which is equivalent to 180°). Therefore, the reference angle for 3π/4 is π/4 (or 45°).
3. **Trigonometric Ratios**: The trigonometric ratios for the reference angle π/4 are:
- sin(π/4) = √2/2
- cos(π/4) = √2/2
- tan(π/4) = 1
Since the angle 3π/4 is located in the second quadrant, we need to adjust the signs of the trigonometric functions:
- sin(3π/4) = sin(π/4) = √2/2 (positive since y-coordinates are positive in the second quadrant)
- cos(3π/4) = -cos(π/4) = -√2/2 (negative since x-coordinates are negative in the second quadrant)
- tan(3π/4) = -tan(π/4) = -1 (negative, as tan is the ratio of sin to cos)
4. **Final Answer**: The exact trigonometric ratios for the angle x = 3π/4 are as follows:
- sin(3π/4) = √2/2
- cos(3π/4) = -√2/2
- tan(3π/4) = -1
If any of the angles were such that their ratio is undefined, we would indicate that with the word “undefined”, but for 3π/4, all the ratios are defined as shown.