Short answer: Yes.
In a right-angled triangle (one of the angles is 90 degrees), a triangle is completely determined by knowing either one side and one other angle, or by knowing two sides. This is because that triangle is similar to all other triangles with the same angle or ratio of sides.
Trigonometry is just a way of standardizing all of the ratios of sides. If we consider one of the angles of a right-angled triangle, and then label the sides hypotenuse (across from the 90 degrees), opposite (across from your angle) and adjacent (side left over), then the all the ratios of sides are given names. Sine is the opposite over hypotenuse, cosine is the adjacent over hypotenuse, tangent is opposite over adjacent, cosecant is hypotenuse over opposite, secant is hypotenuse over adjacent and cotangent is adjacent over opposite.
These are abbreviated sin, cos, tan, csc, sec and cot, but we often just use the first three. Because all of these triangles are similar, it means that we can solve problems very easily (with a little bit of practice first) using our calculator, which is really just a fast way of using reference triangles.