Fundamental and tangent relations

The fundamental relationship

Since the point M(cos(theta), sin(theta)) lies on the circle of radius 1 centred at O, Pythagoras’ theorem gives the fundamental relationship in trigonometry:

cos²(theta) + sin²(theta) = 1

(we write cos²(theta) to mean (cos(theta))²)

This relationship holds true for any angle theta, and it allows us to find cos(theta) if we know sin(theta), or vice versa (up to the sign).

Example

If sin(theta) = 3/5 and theta lies in the first quadrant (positive cos), then: cos^2(theta) = 1 - sin^2(theta) = 1 - 9/25 = 16/25 cos(theta) = sqrt(16/25) = 4/5 (positive because it is in the first quadrant)

The tangent

We also define the tangent of theta (when cos(theta) is not zero):

tan(theta) = sin(theta) / cos(theta)

Geometrically, tan(theta) corresponds to the y-coordinate of the point of intersection between the line (OM) and the vertical line tangent to the circle at point A(1,0).

Example

tan(pi/4) = sin(pi/4) / cos(pi/4) = (sqrt(2)/2) / (sqrt(2)/2) = 1

Summary table

Angle	tan
0	0
pi/6	sqrt(3)/3
pi/4	1
pi/3	sqrt(3)
pi/2	undefined

Common pitfall

The tangent is undefined when cos(theta) = 0, i.e. for theta = pi/2 + k x pi. Never divide by a cosine of zero, and always check the sign of the cosine before ‘taking the square root’ in the fundamental relation.