Tangent Ratio
Opposite over adjacent.
Key idea
Opposite over adjacent.
In depth
tan θ = opposite / adjacent = sin θ / cos θ. It is the slope of the line at angle θ. It grows without bound as θ approaches 90° because the adjacent side shrinks to zero.
Worked example
tan 45° = 1 since opposite and adjacent are equal. A ramp rising 1 for every 4 along has tan θ = 0.25, so θ ≈ 14°.
Formula
Real-world uses
- Measuring the height of a building from a known distance and angle
- Road gradients
Common mistake
Forgetting tan 90° is undefined.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Tangent Ratio?
Opposite over adjacent. It is a topic at the intermediate level in Trigonometry.
What is the formula for Tangent Ratio?
The key formula is tanθ = opp/adj.
Where is Tangent Ratio used in real life?
Real-world examples: Measuring the height of a building from a known distance and angle; Road gradients.
What should I learn before Tangent Ratio?
Make sure you are comfortable with Cosine Ratio, the previous topic in Trigonometry.
What is a common mistake with Tangent Ratio?
Forgetting tan 90° is undefined.
How can I practise Tangent Ratio?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Tangent Ratio?
Move on to SOH-CAH-TOA next.
Related intermediate topics in Trigonometry
- Radians – Angle measured by arc length on a unit circle; 2π rad = 360°.
- Converting Degrees and Radians – Multiply by π/180 or 180/π.
- Sine Ratio – Opposite over hypotenuse.
- Cosine Ratio – Adjacent over hypotenuse.
- Reciprocal Trig Functions – Cosecant, secant and cotangent.
- Solving Right Triangles – Find all sides and angles using ratios and the Pythagorean theorem.