The angle x is given in radians as 3π/4. We need to find the exact trigonometric ratios: sine, cosine, and tangent for this angle.
Sine Ratio
The sine of an angle in the unit circle corresponds to the y-coordinate of the point at that angle. For 3π/4, this angle is located in the second quadrant where sine values are positive.
Since 3π/4 is equivalent to π – π/4, we can use the reference angle of π/4 which has coordinates \\( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \\).
Thus, the sine is:
sin(3π/4) = sin(π - π/4) = sin(π/4) = \frac{\sqrt{2}}{2}Cosine Ratio
The cosine of the angle corresponds to the x-coordinate of the point on the unit circle. In the second quadrant, cosine values are negative.
Using the reference angle π/4:
cos(3π/4) = cos(π - π/4) = -cos(π/4) = -\frac{\sqrt{2}}{2}Tangent Ratio
The tangent of an angle is the ratio of the sine to the cosine. Since we now have:
- sin(3π/4) = \frac{\sqrt{2}}{2}
- cos(3π/4) = -\frac{\sqrt{2}}{2}
We can calculate:
tan(3π/4) = \frac{sin(3π/4)}{cos(3π/4)} = \frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = -1Conclusion
In summary, the exact trigonometric ratios for the angle 3π/4 are:
- sin(3π/4) = \frac{\sqrt{2}}{2}
- cos(3π/4) = -\frac{\sqrt{2}}{2}
- tan(3π/4) = -1