Tangent Ratio (TOA)
Ratio of the opposite side to the adjacent side.
Formula
Variables
Example
tan 45° = 1
How it works
The tangent of an angle in a right triangle tells you how steep the triangle is by comparing the side opposite the angle to the side next to it. It works because similar right triangles keep the same shape, so the ratio of opposite to adjacent stays the same for a given angle. This makes tangent useful whenever you know one leg and need to find the other, or when you want to measure an angle from side lengths.
When to use it
Use tangent in right-triangle problems when you know the opposite and adjacent sides, or when you need to find one of them from an angle. It is especially useful in surveying, navigation, slopes, ramps, and any situation involving height and horizontal distance.
Worked example, step by step
A right triangle has an angle of 30°. The side opposite the 30° angle is 5 cm, and the adjacent side is 8.66 cm. Use the formula: tan θ = opposite / adjacent Substitute the values: tan 30° = 5 / 8.66 Calculate the ratio: 5 / 8.66 ≈ 0.577 So, tan 30° ≈ 0.577
Common mistakes to avoid
- Mixing up the opposite side with the hypotenuse
- Using tangent on non-right triangles without first using a suitable trigonometric method
- Forgetting that the opposite and adjacent sides depend on which angle you are measuring
Did You Know?
Tangent gets its name from the tangent line to a circle, where its length physically represents the ratio.
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More in Trigonometry
View allSine Ratio (SOH)
BasicRatio of the side opposite an angle to the hypotenuse.
Cosine Ratio (CAH)
BasicRatio of the side adjacent to an angle to the hypotenuse.
Cosecant
IntermediateReciprocal of sine.
Secant
IntermediateReciprocal of cosine.
Cotangent
IntermediateReciprocal of tangent.
Tangent Identity
BasicTangent equals sine divided by cosine.