The tangent ratio
Objective. Connect an angle to opposite and adjacent sides.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
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Learn: the key idea
The tangent ratio connects an angle in a right-angled triangle to the lengths of the side opposite that angle and the side adjacent to it, defined as tan(θ) = opposite ÷ adjacent. This ratio is constant for any given angle, regardless of the triangle's overall size, which is why it can be used to find missing sides or angles. For example, in a right-angled triangle where the angle is 30° with an adjacent side of 10 cm, the opposite side equals 10 × tan(30°). The tangent ratio only involves the two sides that are not the hypotenuse, distinguishing it from sine and cosine.
Worked example
In a right-angled triangle, the angle is 40° and the adjacent side is 9 cm. Using the tangent ratio, set up the calculation for the opposite side.
- 1The tangent ratio states tan(θ) = opposite ÷ adjacent.
- 2Rearrange to find the opposite side: opposite = adjacent × tan(θ).
- 3Substitute the known values: opposite = 9 × tan(40°).
- 4Using tan(40°) ≈ 0.839, opposite ≈ 9 × 0.839 ≈ 7.55 cm.
Answer: Approximately 7.55 cm
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Practice
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Start practisingCommon mistake. Pupils sometimes confuse which side is opposite and which is adjacent relative to the marked angle, swapping them in the ratio and getting a reciprocal value by mistake. Always identify the angle first, then label opposite and adjacent relative to that specific angle.
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