How do you determine the exact value of cos 45°?

The exact value of cos 45° is \(\frac{\sqrt{2}}{2}\).

To determine the exact value of cos 45°, we can use the properties of a 45°-45°-90° triangle. This type of triangle is isosceles, meaning it has two equal sides and two equal angles. In a 45°-45°-90° triangle, the two non-hypotenuse sides are equal in length. If we assume each of these sides has a length of 1, we can use the Pythagorean theorem to find the length of the hypotenuse.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. For our triangle, this means:
\[ \text{Hypotenuse}^2 = 1^2 + 1^2 = 1 + 1 = 2 \]
So, the hypotenuse is:
\[ \text{Hypotenuse} = \sqrt{2} \]

The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. For a 45° angle in our triangle, the adjacent side is 1 and the hypotenuse is \(\sqrt{2}\). Therefore:
\[ \cos 45° = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}} \]

To rationalise the denominator, we multiply the numerator and the denominator by \(\sqrt{2}\):
\[ \cos 45° = \frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{\sqrt{2}}{2} \]

Thus, the exact value of cos 45° is \(\frac{\sqrt{2}}{2}\).