Fundamental relationships: cosine, sine and tangent

The fundamental relationship

For any angle x, the point (cos(x), sin(x)) lies on the unit circle, so we always have:

cos²(x) + sin²(x) = 1

This identity forms the basis for many simplifications. For example, if cos(x) = 3/5 and x lies in the first quadrant, then sin²(x) = 1 – 9/25 = 16/25, so sin(x) = 4/5 (positive because x lies in the first quadrant).

The tangent

The tangent is defined by tan(x) = sin(x)/cos(x), provided that cos(x) is not equal to 0 (i.e. x is not equal to π/2 + kπ).

By dividing the fundamental relation by cos²(x), we obtain a useful formula:

1 + tan²(x) = 1/cos²(x)

Sign table

Quadrant	cos(x)	sin(x)	tan(x)
1 (0 to π/2)	+	+	+
2 (π/2 to π)	-	+	-
3 (π to 3π/2)	-	-	+
4 (3π/2 to 2π)	+	-	-

Common pitfall

The sign of sin(x) or cos(x) depends on the quadrant: sqrt(1 - cos²(x)) always gives a positive value, so remember to add a negative sign if sin(x) is required to be negative in the context of the exercise.