Sine, cosine and tangent

At school we first learn how to measure angles and how to add and subtract angles. Later on comes the idea of sine, cosine and tangent. There is a question that is left unanswered until we learn the definitions of sine, cosine and tangent. The question is: we know that a ramp can be harder to walk on according to how much inclined it is, with flat being zero angle and 90° the hardest. Is there a relationship of how many units we walk forwards and how many units we go up if we are walking over a ramp with an angle greater than zero? This is exactly what sine, cosine and tangent are. Ratios that relate angles to sides of a triangle such that we have the answer for the previous mentioned question.

If you understand how to read the unit circle the trigonometric identities are all consequences of it. There are multiple formulas regarding the product, the quotient, addition and subtraction of angles. I have a textbook that states that as long as you understand the sum of angles, it's really all you need to know. I'm going with that textbook and going to show just two identities.