Answer:
The acute angle that has got the same cosine of the angle and cosine of the angle is a 45° angle.
Explanation:
Two acute angles have got the same sine of the angle and cosine of the angle if they are complementary angles.
Two acute angles α and β are complementary angles if adding both of them results a rectangle angle (α + β = 90°).
If we draw a rectangle with vertices A B C D, and then we divide it in two triangles by tracing a diagonal line matching A vertice with B vertice, we can obtain two triangles.
The diagonal would fulfill the function of the hypotenuse of both of the triangles.
If we take the A vertice we can observe that there are two complementary angles at both sides of the diagonal.
We can call them α and β.
α + β = 90°
sine α = opposite side / hypotenuse
cosine β = adjacent side / hypotenuse
But we can observe that the opposite side of one triangle has the same length as the adjacent side of the the other, and the hypotenuse is the same in both triangles, therefore sine α = cosine β.
Examples:
α=30°; β=60°
α + β= 90°
sine α = sine 30° = 0,5
cosine β = cosine 60° = 0,5.
The only one case in which the same angle has the same sine and cosine at the same time is when α = β = 45°. Both α and β are complementary angles and they measure the same amplitude.
α=45°; β=45°
α + β= 90°
sine α = sine 45° = √2/2
cosine β = cosine 45° = √2/2