The Quotient Identity
Simple Explanation
The tangent of any angle is always equal to its sine divided by its cosine: tan θ = sin θ / cos θ.
Why Do We Need It?
This connects all three main trigonometric ratios into a single relationship, letting you find tan θ whenever sin θ and cos θ are already known, without a separate calculation.
Formula
The Quotient Identity
tan θ = sin θ / cos θ
The tangent of an angle is always equal to its sine divided by its cosine.
- θ
- — any angle for which cos θ ≠ 0
When to use it: Whenever tan θ needs to be found from known sin θ and cos θ values, or vice versa.
Worked Example
Find tan θ from sin θ and cos θ
Given sin θ = 0.6 and cos θ = 0.8, find tan θ using the quotient identity.
Why Does This Work?
From the general definition, sin θ=y/r and cos θ=x/r. Dividing sin θ by cos θ gives (y/r)/(x/r) = y/x — and y/x is exactly the definition of tan θ, so the identity follows directly.
Real-Life Example
Finding a slope angle's tangent from its components
A surveyor has already measured a slope's vertical and horizontal components (essentially sine and cosine of the slope angle) separately.
The quotient identity gives the slope's tangent (its gradient) directly, without re-measuring anything new.
Practice
Given sin θ = 0.8 and cos θ = 0.6, find tan θ. (Give your answer to 2 decimal places.)
MediumCommon mistake
Inverting the quotient (computing cos θ/sin θ instead of sin θ/cos θ) — that reversed ratio is a different quantity (cotangent), not tangent.
Quick Review
- tan θ = sin θ / cos θ, for any angle where cos θ ≠ 0.
- Follows directly from the general definitions: (y/r)/(x/r) = y/x.
- Lets you find tan θ instantly once sin θ and cos θ are known.