Meaning of π :
No matter how big or small a circle, or the units used to measure the circle's circumference and diameter, the ratio of the circle's circumference to its diameter is always the same number. The symbol π is the tag/label we use to identify the ratio of the circumference of a circle to its diameter.
The tags "sin", "cos", and "tan" are also symbols (albeit three letter symbols) used identify ratio of the sides of a triangle.
Meaning of "sin", "cos", "tan"
The ratio of two sides of any right triangle with the same interior angles is always the same number independent of the size of the triangle, the triangle's orientation, or the units used to measure the sides of the triangles. See similar right triangles for some specific example.
There are three ways to form a ratio of the three sides of a triangle (six if you count the inverse ratios also). The symbols "sin", "cos", and "tan" are the tags/labels we use to identify which ratio is which.
Angles of Right Triangles:
- A right triangle is any triangle in which one of its angles is 90o.
- In any right triangle there are two angles that one could focus attention on besides the right angle.
- The sum of these two angles is always equal to 90o, θ + φ = 90o. If you know one of the angles you can always find the other since it is the complement of the known angle, i.e. the other angle is equal to 90o minus the known angle.
- The side of the triangle that is the called the Hypotennuse is always the side opposite the right angle.
- When determining the sine, cosine, or the tangent of an angle, the Opposite and Adjacent sides for this angle is not the same as the other complementary angle. In fact they are the opposite of each other.
The side opposite the right angle
Opposite Side: opp
The side opposite the angle of focus.
Adjacent Side: adj
The side connected to the angle of focus.
Trigonometric Relations for either Angle:
- If one chooses an angle first and labels the sides relative to that angle, then
- The advantage of this method is that it works regardless of the orientation of the triangle or the labels used for the angles - there is nothing sacrosanct about the labels q and f.