To solve this problem, let's go back to the basic unit circle.
Refer to the diagram. We know that the vertical component of the triangles is simply the sine of the angle since we can recall that sine = opposite/hypotenuse (from SOHCAHTOA) and the hypotenuse is equal to the radius of the UNIT circle which is equal to 1. Correspondingly, the horizontal component of the triangle will be the cosine of the angle.
The figure on the left shows the 20-degree angle. We know that the side opposite it measures 0.342. Now, let's look at the figure on the right.
Note that the angle on the topmost would be equal to 20 degrees since the angles below are 70 and 90 degrees respectively. Since it's also equal to 20 degrees, then we know that the side opposite it measures 0.342 too! This side also happened to be the cosine of 70. This is the reason why the sine of 20 is equal to the cosine of 70.
Generally speaking, this reasoning will work for ANY pairs of complementary angles. The sine of an angle is always equal to the cosine of its complementary angle.
