1. The problem is to find the values of $\sin 45^\circ$, $\cos 45^\circ$, and $\tan 45^\circ$. 2. Recall the definitions and important values for these trigonometric functions at $45^\circ$: - $\sin \theta$ is the ratio of the opposite side to the hypotenuse in a right triangle. - $\cos \theta$ is the ratio of the adjacent side to the hypotenuse. - $\tan \theta$ is the ratio of the opposite side to the adjacent side, or $\tan \theta = \frac{\sin \theta}{\cos \theta}$. 3. For $45^\circ$, the right triangle is isosceles with legs of equal length. If each leg is 1, the hypotenuse is $\sqrt{1^2 + 1^2} = \sqrt{2}$. 4. Calculate each function: - $\sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ after rationalizing the denominator. - $\cos 45^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$. - $\tan 45^\circ = \frac{\sin 45^\circ}{\cos 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$. 5. Final answers: $$\sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2}, \quad \tan 45^\circ = 1.$$