To solve this problem, we start by drawing a right-angled triangle with one angle of 45°. In a right-angled triangle, the two non-right angles must sum to 90°. Therefore, if one of these angles is 45°, the other must also be 45°. This makes the triangle an isosceles right-angled triangle, where the two legs are of equal length.

Given that one of the shorter sides (one leg of the triangle) has a length of 1, we can determine the lengths of the other sides as follows:

1. Both legs are equal in length: Since the triangle is isosceles and one leg is 1, the other leg is also 1.

2. Hypotenuse calculation: The length of the hypotenuse (let's denote it as $c$) can be found using the Pythagorean theorem, which states:

   $c^2 = a^2 + b^2$

   Substituting the known lengths:

   $c^2 = 1^2 + 1^2 = 1 + 1 = 2$

   Thus, $c = \sqrt{2}$.

Now, we can find the exact values of the trigonometric functions for the 45° angle:

1. $\sin 45^\circ$: $\sin 45^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}$

2. $\cos 45^\circ$: $\cos 45^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}$

3. $\tan 45^\circ$: $\tan 45^\circ = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{1} = 1$

For exact values, we often rationalize the denominators:

• $\sin 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
• $\cos 45^\circ = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$
• $\tan 45^\circ = 1$

Thus, the exact values are:

• $\sin 45^\circ = \frac{\sqrt{2}}{2}$
• $\cos 45^\circ = \frac{\sqrt{2}}{2}$
• $\tan 45^\circ = 1$

Would you like any further explanation or have any questions about these results?

Here are 5 questions you might consider asking next:

1. How can we use trigonometry to find the area of the triangle?
2. What are the properties of isosceles right-angled triangles?
3. Can you explain why the sine and cosine of 45° are equal?
4. How do you rationalize a fraction with a square root in the denominator?
5. What are some real-life applications of right-angled triangles?

Tip: In trigonometry, rationalizing denominators is often done to simplify expressions, making them easier to work with or recognize.