Use of Identity tan θ = sin θ / cos θ

Understanding the Identity tan θ = sin θ / cos θ

This equation is considered as a fundamental trigonometric identity, which defines tangent (tan) of an angle as the ratio of sine (sin) to cosine (cos) for the same angle.
Just like the sine and cosine, the tangent function is also periodic with a period of π or 180°, that is, tan(θ + π) = tan θ for any angle θ.
Tangent function has asymptotes at odd multiples of π/2 (or 90°), which are the zeros of the cosine function. Therefore, the identity is not valid when cos θ = 0.

Applying the Identity

The identity tan θ = sin θ / cos θ can be used to find the tangent of an angle if the sine and cosine for that angle are known.
For instance, if for a certain angle θ, sin θ = 0.6 and cos θ = 0.8, then, tan θ can be found by taking their ratio, so tan θ = 0.6 / 0.8 = 0.75.
Inversely, if tan θ and either sin θ or cos θ are known, this identity can be rearranged to find the other.

Important Things to Note

Trigonometric identities, including the tan θ = sin θ / cos θ, prove to be invaluable for solving trigonometric equations or simplifying expressions.
The tangent function changes sign in each quadrant. Hence, the sign of tan θ depends on which quadrant the angle θ is in.
Division by zero is undefined in mathematics. Therefore, ensure not to use the identity when cos θ = 0.
For any angles in the form (2n + 1) * π / 2, the tangent is undefined because the cosine of these angles is zero. Therefore, the solutions of trigonometric equations involving tangent should never include angles of this form.