Chapter 11.1 — Introduction to Trigonometry

Opposite, Adjacent, Hypotenuse

These three labels are relative to whichever angle is currently being measured, not fixed to a particular side. Relative to ∠C in the same triangle, the roles of the two shorter sides swap — AB becomes opposite, BC becomes adjacent — while the hypotenuse, AC, stays the hypotenuse regardless of which acute angle is under discussion, since it's always the side opposite the right angle itself. Getting this relabelling right before writing down any ratio is worth doing deliberately, since the single most common trigonometry mistake is naming a side "opposite" or "adjacent" without first checking which angle the problem is actually asking about.

Why the Ratios Never Change

Scale a right triangle up or down, and its angles stay exactly identical — which means every ratio between its sides stays identical too, no matter how much bigger or smaller the triangle gets. Four nested similar triangles, each a scaled copy of a 3-4-5 right triangle, make this concrete:

Every triangle here is a different size, yet both ratio columns land on the exact same value every single time. That fixed value — 4/5 for opposite-over-hypotenuse, 3/5 for adjacent-over-hypotenuse — depends only on the angle, never on how large the triangle happens to be. These two constant ratios earn their own names: opposite-over-hypotenuse is called sine, adjacent-over-hypotenuse is called cosine, and opposite-over-adjacent is called tangent.

This is really the entire foundation the whole chapter is built on: since a ratio depends only on the angle, it can be looked up or computed once for a given angle and then reused in any triangle containing that same angle, no matter how big or small that particular triangle is. Without this constancy, trigonometry wouldn't work as a general tool at all — every new triangle would need its own fresh measurement, with nothing carrying over from one problem to the next, and no such thing as a reusable reference table of standard angle values would even be possible.

All Six Ratios, Defined

The last three ratios above are simply the first three flipped upside down — cosecant is the reciprocal of sine, secant the reciprocal of cosine, cotangent the reciprocal of tangent. Six names exist purely for convenience; only two ratios are genuinely independent, since every other ratio can be built from sine and cosine alone (tanA=sinA/cosA, and each of the other three is a straight reciprocal).

Two reciprocal-pair facts fall out immediately and are worth stating on their own: sinA·cosecA=1 always, and cosA·secA=1 always, and tanA·cotA=1 always, simply because a number multiplied by its own reciprocal is always exactly 1. These aren't separate rules to memorise — they're the direct consequence of how cosecant, secant, and cotangent were defined in the first place, flipped versions of sine, cosine, and tangent respectively.

Finding All Six Ratios From Two Sides

Given AC=13 cm (hypotenuse) and BC=5 cm in a right triangle, the third side comes from Pythagoras before any ratio can even be found at all.

Every single trigonometric-ratio problem in this chapter follows this identical two-step shape: recover the missing side with Pythagoras first, then simply read off whichever ratios the question actually asks for. It's worth noticing, too, that these two steps use two completely different tools: Pythagoras' theorem finds a length, using nothing but the two other lengths already known, while the ratio definitions themselves are what turn those three lengths into the actual trigonometric values being asked for.

A second, slightly different example makes the pattern even clearer: in a right triangle with legs 7 cm and 24 cm meeting at the right angle, the hypotenuse is √(49+576)=√625=25 cm — another clean 7-24-25 triple. From there, sine and cosine of either acute angle drop straight out of the three now-known side lengths, exactly the same way as before. Recognising a Pythagorean triple like 3-4-5, 7-24-25, or 5-12-13 the moment two of the three sides are given is a genuinely useful shortcut throughout this whole chapter, since it means the third side can often be found by inspection rather than by working through the full squaring-and-subtracting calculation every single time.

Four Quick Conceptual Checks

• Can sinθ ever equal 4/3? No — sine is always a proper fraction, a leg divided by the hypotenuse, and the hypotenuse is always the longest side of a right triangle. A ratio greater than 1 can never be a valid sine or cosine value, which is a genuinely useful quick sanity check on any computed answer: if a sine or cosine ever comes out greater than 1, something upstream in the working has gone wrong.
• Why are sinA and cosA always less than 1? Because both are built from a leg divided by the hypotenuse, and a leg is always strictly shorter than the hypotenuse in any right triangle — the hypotenuse is, after all, the longest side of any right triangle by definition, opposite the largest angle.
• Is tanA the product of "tan" and "A"? No — tanA is one indivisible symbol, the ratio of opposite to adjacent side. It can never be split apart into a separate "tan" multiplied by a separate "A," the same way "log x" can never be split into "log" multiplied by "x" in any other branch of mathematics.
• Is sinA/cosA the same as tanA? Yes — substituting the definitions directly, (BC/AC)/(AB/AC) simplifies by cancelling the shared AC, leaving BC/AB, which is exactly tanA. The identical logic run the other way round shows cosA/sinA equals cotA, since dividing adjacent-over-hypotenuse by opposite-over-hypotenuse cancels the same shared AC and leaves AB/BC — the definition of cotangent.

Where This Chapter Goes

Exercise 11.1 practises finding all six ratios from two given sides. Exercise 11.2 derives exact ratio values for five special angles — 0°, 30°, 45°, 60°, 90°. Exercise 11.3 connects an angle's ratios to its complement's. Exercise 11.4 closes the chapter with algebraic identities built entirely on Pythagoras' theorem — the same theorem this introduction already leaned on twice, now reappearing in a purely algebraic form rather than a numeric one.