Answer:
The Pythagorean Theorem states that, for a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Therefore, if we have a right triangle with angles A, B, and C, then:
$$A^2 + B^2 = C^2$$
Now, let's consider a triangle with angles of 75°, 45°, and 30°. We know that the sum of the angles must equal 180°, so these three angles add up to 180°.
We can now plug in the angles into the formula above and solve for the hypotenuse C:
$$(45^2) + (30^2) = C^2$$
$$2025 = C^2$$
$$\sqrt{2025} = C$$
Therefore, the length of the hypotenuse C is 45 units.
Now, let's look at the cosine of angle A, which is 75°. By definition, the cosine of an angle is equal to the length of the adjacent side divided by the length of the hypotenuse. Therefore, for this triangle:
$$\cos 75° = \frac{\text{Adjacent side}}{\text{Hypotenuse}} = \frac{\text{Adjacent side}}{45}$$
We can now find the length of the adjacent side by subtracting the length of the opposite side from the hypotenuse:
$$\text{Adjacent side} = 45 - \text{Opposite side}$$
The length of the opposite side is equal to the length of the other leg, which is 30 units in this case. Therefore, the length of the adjacent side is 15 units.
Putting this all together, we can now solve for the cosine of 75°:
$$\cos 75° = \frac{15}{45} = \frac{\sqrt{2} (\sqrt{3}-1)}{4}$$
Therefore, cos 75° = √2 (√3-1)÷4.