Geometric Interpretation

 

The trigonometric functions are defined from the orthogonal triangular as a ratio of its sides. In order to interpret them geometrically we will use the unit circle. The unit circle consists of a circle of radius length of one unit, which is set on the centre of the Cartesian coordinates. We will also consider a line of angle θ in respect to axis x, that begins from the origins of the Cartesian system and intersects the unit circle at the point A.

In image 1 we can see how sine, cosine and tangent are geometrically interpreted. Sine is the length of the y axis from its origin up to the projection of the point A on the y axis, cosine is the length of the x axis from its origin up to the projection of the point A on the x axis and tangent is the length of the tangent of the unit circle at the point A, from that point, up to the point that it intersects the axis x.

 

Image 1: Sine, Cosine, Tangent

 

In image 2 we can see how secant, cosecant and cotangent are geometrically interpreted. Secant is the length of the x axis from its origin up to the point where the tangent of point A intersects the x axis. Cosecant is the length of the y axis from its origin up to the point where the tangent of point A intersects the y axis. Cotangent is the length of the tangent of point A, from this point, up to the point that it intersects the y axis.

 

 

Image 2: Secant, Cosecant, Cotangent

 

Angle θ can vary around the unit circle. This is why the graphs repeat themselves every 360 degrees.