1. **Problem:** Find the exact values of the six trigonometric functions for the angle $\frac{7\pi}{4}$ radians. 2. **Recall the six trigonometric functions:** - $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ - $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$ - $\tan \theta = \frac{\sin \theta}{\cos \theta}$ - $\csc \theta = \frac{1}{\sin \theta}$ - $\sec \theta = \frac{1}{\cos \theta}$ - $\cot \theta = \frac{1}{\tan \theta}$ 3. **Locate the angle $\frac{7\pi}{4}$:** - $\frac{7\pi}{4} = 2\pi - \frac{\pi}{4}$, so it is in the fourth quadrant. - Reference angle is $\frac{\pi}{4}$. 4. **Use known values for $\frac{\pi}{4}$:** - $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ - $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ - $\tan \frac{\pi}{4} = 1$ 5. **Apply signs in the fourth quadrant:** - $\sin$ is negative - $\cos$ is positive - $\tan$ is negative 6. **Calculate each function:** $$\sin \frac{7\pi}{4} = -\frac{\sqrt{2}}{2}$$ $$\cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2}$$ $$\tan \frac{7\pi}{4} = -1$$ $$\csc \frac{7\pi}{4} = \frac{1}{\sin \frac{7\pi}{4}} = \frac{1}{-\frac{\sqrt{2}}{2}} = -\frac{2}{\sqrt{2}} = -\sqrt{2}$$ $$\sec \frac{7\pi}{4} = \frac{1}{\cos \frac{7\pi}{4}} = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$$ $$\cot \frac{7\pi}{4} = \frac{1}{\tan \frac{7\pi}{4}} = \frac{1}{-1} = -1$$ **Final answers:** $$\sin \frac{7\pi}{4} = -\frac{\sqrt{2}}{2}, \quad \cos \frac{7\pi}{4} = \frac{\sqrt{2}}{2}, \quad \tan \frac{7\pi}{4} = -1,$$ $$\csc \frac{7\pi}{4} = -\sqrt{2}, \quad \sec \frac{7\pi}{4} = \sqrt{2}, \quad \cot \frac{7\pi}{4} = -1.$$