Ratios and Graphs of Trigonometric Functions · Form 5
What do sine, cosine and tangent actually mean?
Sine, cosine and tangent are ratios that compare two sides of a right-angled triangle for a chosen angle, sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent. Their value depends only on the angle, not on how big the triangle is.
A ratio, not a length
Each one is a comparison, written as a fraction of two sides measured against the angle you are looking at: sine pairs the opposite side with the hypotenuse, cosine pairs the adjacent side with the hypotenuse, and tangent pairs the opposite side with the adjacent side. Many students remember this as SOH-CAH-TOA.
Because you are dividing a length by a length, the answer is a plain number with no units, such as sin 30° = 0.5.
Why the same angle always gives the same value
All right-angled triangles that share the same angle are similar, so their matching sides are always in the same proportion. That is why sin 30° stays 0.5 whether the triangle is drawn tiny or as big as a football field.
This fixed link between an angle and its ratio is what lets a calculator store one value for each angle, and it is what makes trigonometry useful for finding a missing side or a missing angle.
The mix-up to watch for
The commonest slip is naming the sides wrongly: 'opposite' and 'adjacent' are decided by the angle you picked, while the hypotenuse is always the longest side, facing the right angle. It also helps to remember that sin θ is one whole quantity, not 'sin times θ' you cannot split it up.
A quick way to recognise a trig-ratio question is that it gives a right-angled triangle with an angle and a side, and asks for another side or angle.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I remember which sides go with sine, cosine and tangent?
Use SOH-CAH-TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. In exam questions, first label the sides relative to the angle you're using, the "opposite" and "adjacent" sides swap if you switch to the other acute angle in the triangle.
Why doesn't the triangle's size change the ratio?
Similar triangles keep the same angles, so their side ratios stay equal even as the triangle grows or shrinks. That's why sin 30° is always 0.5, whatever the hypotenuse length, the ratio depends only on the angle, letting you use a small table of values for any triangle.
What mistake do students often make with opposite and adjacent sides?
They label the sides using the wrong angle. In a right-angled triangle, "opposite" and "adjacent" are defined relative to the marked angle, not fixed positions, if the question asks for the other acute angle's ratios, opposite and adjacent switch places, so always check which angle is being used first.