Trigonometric Ratios: sin, cos, tan

1Summary 重點摘要

In a right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. Remember SOH-CAH-TOA. The key is labelling 'opposite' and 'adjacent' from the reference angle — opposite is across from it, adjacent is next to it (not the hypotenuse).

2Vocabulary cards Vocabulary

3Step by step Steps

1. Trig ratios and SOHCAHTOATrigonometric ratios describe proportions between sides of a right-angled triangle: sine, cosine and tangent. The mnemonic SOHCAHTOA gives sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, and tan θ = opposite ÷ adjacent. The ratios depend only on the angle, not on the size of the triangle: doubling the triangle doubles every side yet leaves each ratio unchanged — which is exactly why trigonometry can measure tower heights and river widths that no tape can reach.
2. Step one: identify the hypotenuseThe hypotenuse is the longest side of a right-angled triangle, always facing the right angle, regardless of which acute angle you choose. In a right triangle with angle θ marked, the hypotenuse is the same side whether θ sits at the lower left or the lower right. The rule: find the right angle first, and the side opposite it is the hypotenuse. Common errors include mistaking an adjacent side for the hypotenuse after the figure is rotated; circling the right angle and labelling the hypotenuse first prevents such slips.
3. Step two: identify the opposite and adjacent sidesThe opposite and adjacent sides depend on the chosen angle: the opposite side faces angle θ, while the adjacent side is the other leg touching θ. With θ at the bottom left, the vertical side on the right faces θ and is the opposite; the horizontal base touches θ and is the adjacent. Choose the other acute angle and the two swap. So always state 'with respect to θ' and label the three sides, marking O, A and H directly on the figure to avoid confusion.
4. Finding sin with SOHUse sine when the opposite side and hypotenuse are involved: sin θ = opposite ÷ hypotenuse. With opposite 3 and hypotenuse 5, sin θ = 3 ÷ 5 = 0.6. The same formula finds a side when the angle is known: opposite = hypotenuse × sin θ. The classic error is swapping opposite with adjacent, which actually computes the cosine; labelling O, A and H before writing the ratio prevents it. On the calculator, confirm the DEG mode first, or every answer will be wrong.
5. Finding cos with CAHUse cosine when the adjacent side and hypotenuse are involved: cos θ = adjacent ÷ hypotenuse. With adjacent 4 and hypotenuse 5, cos θ = 4 ÷ 5 = 0.8. Rearrangements give adjacent = hypotenuse × cos θ and hypotenuse = adjacent ÷ cos θ. For acute angles the cosine ratio always lies between 0 and 1; an answer above 1 signals a wrong side or an inverted division and demands an immediate re-check — never write down an impossible value.
6. Finding tan with TOA and choosing the ratioUse tangent when the two legs are involved: tan θ = opposite ÷ adjacent. With opposite 3 and adjacent 4, tan θ = 3 ÷ 4 = 0.75; among the three ratios only tan may exceed 1. In a problem, ask first which sides are given, then pick the ratio that uses them while leaving exactly one unknown; if both legs are known and the hypotenuse is wanted, Pythagoras' theorem works too. The guiding rule is: use all the givens and keep a single unknown — choose well and the calculation flows.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• SOH: sin = opp / hyp
• CAH: cos = adj / hyp
• TOA: tan = opp / adj
• Hypotenuse is always the longest side
• Opposite and adjacent depend on where angle θ is