To understand the relationship between the cosine of 30 degrees and the sine of 60 degrees, we can draw upon fundamental trigonometric identities and properties of the angles in a right triangle.
We start with the given information: the cosine of 30 degrees is indeed √3/2. This is a well-known value in trigonometry that comes from the unit circle or the special triangles. Specifically, in a 30-60-90 triangle, the sides have the following ratios:
- The side opposite the 30-degree angle (the shortest side) is 1.
- The side opposite the 60-degree angle is √3.
- The hypotenuse (the longest side) is 2.
From this triangle, we can derive the values of the sine and cosine for both angles. The cosine function relates to the adjacent side over the hypotenuse, while the sine function relates to the opposite side over the hypotenuse. Therefore:
For cosine:
- cos(30°) = adjacent/hypotenuse = √3/2
And for sine:
- sin(60°) = opposite/hypotenuse = √3/2
This means that the sine of 60 degrees is equal to the cosine of 30 degrees. Thus, we can conclude:
sin(60°) = √3/2.
In summary, since we’ve established that cosine of 30 degrees equals √3/2, we can confidently state that the sine of 60 degrees also equals √3/2. This beautifully illustrates the symmetry in trigonometric functions concerning complementary angles.