Unit Circle Definition of Sine and Cosine Functions
The trigonometric functions can be defined in terms of a unit circle, i.e. a circle of radius one.
The sin/cos Triangle
If the unit circle is placed at the origin of a rectangular coordinate system with the angle q measured from the positive x-axis to the terminal side, then the point on the unit circle where the terminal side intersects the unit circle is defined to be (cos q, sin q), i.e. the first coordinate of a point on the unit circle is the cos q and the second coordinate is sin q.
The tan/sec Triangle
The tan q and sec q are defined by a triangle whose height is tangent to the unit circle at the point (1, 0) and whose hypotenuse is on the terminal side of the angle.
The cot/csc Triangle
The cot q and csc q are defined by a triangle whose height is one and whose hypotenuse is on the terminal side of the angle.
The sin/cos, tan/sec, and cot/csc Triangles
All three of the triangles used to define the trigonometric functions are shown in figure below.
Using the sin/cos, tan/sec, and cot/csc Triangles to Establish Basic Trigonometric Identites
The three similar triangles sin/cos, tan/sec, and cot/csc are extracted from the figure. A fourth similar triangle is shown with the adjacent, opposite, and hypotenuse sides labelled.
The definition of the six trigonometric functions and other useful identities follow from using the fact that the ratio of corresponding sides of similar triangles must be the equal. The results are:
Using the sin/cos, tan/sec, and cot/csc Triangles to Determine the Pythagorean Idenities
The Pythagorean Theorem states:
In any right-angled triangle, the sum of the squares of the lengths of the sides containing the right angle is equal to the square of the hypothenuse.
In short c2 = a2 + b2.
Applying the Pythagorean Theorem to the sin/cos, tan/sec, and cot/csc triangles gives: