Hello, and welcome to your Class 9 Mathematics lesson. Today, we begin an exciting new chapter: Trigonometrical Ratios. Over the next few minutes, we will explore how the sides and angles of right-angled triangles are connected through special ratios. By the end, you will understand sine, cosine, tangent, and their reciprocal ratios — and you'll see how to apply them to solve problems.
Let us start with the word itself. Trigonometry comes from Greek roots meaning "measurement of triangles." In this chapter, we focus only on right-angled triangles — triangles with one angle exactly equal to 90°.
Imagine a right-angled triangle. Pick any one of the two acute angles — this becomes your angle of reference. Now, look carefully at the three sides relative to this angle. The side directly opposite your angle of reference is called the perpendicular. The side that lies next to your angle, forming one of its arms, is called the base. The longest side, always opposite the right angle, is called the hypotenuse.
Here is the crucial point: these labels change depending on which angle you choose as your reference. If you switch to the other acute angle, what was perpendicular becomes base, and vice versa. But the hypotenuse always stays the same — it is always the side opposite the right angle.
In trigonometry, we often use Greek letters for angles. The most common is θ, but you will also encounter φ, α, β, and γ. Do not be intimidated by these symbols — they are simply names for angles, just like ∠A or ∠C.
Now we come to the heart of this chapter: the six trigonometrical ratios. These are ratios formed by comparing pairs of sides in a right-angled triangle.
Consider a right-angled triangle ABC, with the right angle at B. Let angle A be our reference angle.
First, sin A is defined as the ratio of perpendicular to hypotenuse. So sin A = BC/AC.
Second, cos A is defined as the ratio of base to hypotenuse. So cos A = AB/AC.
Third, tan A is defined as the ratio of perpendicular to base. So tan A = BC/AB.
These three ratios have reciprocals, giving us three more ratios.
cot A is base over perpendicular, the reciprocal of tangent. cot A = AB/BC.
sec A is hypotenuse over base, the reciprocal of cosine. sec A = AC/AB.
cosec A is hypotenuse over perpendicular, the reciprocal of sine. cosec A = AC/BC.
Remember this: every trigonometrical ratio is a pure number. It has no units, because you are dividing one length by another. The units cancel out.
Let us work through an example to see these ratios in action.
Suppose in right-angled triangle ABC, with right angle at B, we have AB = 5 and AC = 13. First, we find the missing side using Pythagoras theorem. Since AC² = AB² + BC², we get 169 = 25 + BC², so BC = 12.
Now, for angle A: the perpendicular is BC equals 12, the base is AB equals 5, and the hypotenuse is AC equals 13. Therefore, sin A = 12/13, cos A = 5/13, and tan A = 12/5.
Notice something interesting: for angle C, the perpendicular becomes AB and the base becomes BC. So sin C = 5/13 and cos C = 12/13. See how sine of one acute angle equals cosine of the other? This is not a coincidence — we will explore such patterns later.
Here is another useful example. Suppose you are told that cot A = 4/3 in a right-angled triangle. This means base over perpendicular equals 4/3.
We can introduce a constant: let base be 4x and perpendicular be 3x. By Pythagoras theorem, hypotenuse squared equals 16x² + 9x², which is 25x². So hypotenuse equals 5x.
Now all ratios follow easily: sin A = 3/5, cos A = 4/5, tan A = 3/4, sec A = 5/4, and cosec A = 5/3. This technique of using a variable multiplier is powerful — use it whenever a ratio is given but not actual lengths.
Let us now examine the reciprocal relationships between these ratios.
Since sin A is perpendicular over hypotenuse, and cosec A is hypotenuse over perpendicular, they are reciprocals. Therefore, sin A = 1/cosec A and cosec A = 1/sin A.
Similarly, cos A and sec A are reciprocals, as are tan A and cot A.
There is also a beautiful connection: tan A = sin A/cos A. This follows directly from the definitions — dividing perpendicular by hypotenuse, then dividing by base over hypotenuse, gives perpendicular over base. Likewise, cot A = cos A/sin A.
Here is a practical problem to bring this together.
Given that 13 sin A = 12, find sec A − tan A.
From the given, sin A = 12/13. So perpendicular is 12x and hypotenuse is 13x for some x. By Pythagoras, base squared equals 169x² − 144x², which is 25x². So base equals 5x.
Therefore, sec A = 13/5 and tan A = 12/5. Their difference is 1/5.
Notice also that sec²A − tan²A equals 169/25 − 144/25, which simplifies to 25/25, or simply 1. This is not accidental — it hints at fundamental identities we will study in detail later.
Let us try one more example involving a diagram with two right triangles.
Suppose in one triangle, cos α = 5/13, and in another, cos β = 3/5. We need to find a total length composed of sides from both triangles.
From the first: if base is 5k and hypotenuse is 13k, then perpendicular is 12k. Using Pythagoras: 169k² = 144 + 25k², giving k = 1. So the first side is 5 units.
From the second: if base is 3p and hypotenuse is 5p, then perpendicular is 4p. Given that this perpendicular equals 8, we get p = 2. So the second side is 6 units.
The total length is therefore 5 plus 6, which equals 11 units. This shows how trigonometry helps us find unknown lengths in complex figures.
Before we conclude, let us recap the essential points you should carry forward.
First, in any right-angled triangle, identify the perpendicular, base, and hypotenuse relative to your chosen acute angle. These labels depend on the angle of reference.
Second, the six trigonometrical ratios are: sine, cosine, tangent, and their reciprocals — cosecant, secant, and cotangent. Know their definitions by heart.
Third, sine and cosecant are reciprocals; cosine and secant are reciprocals; tangent and cotangent are reciprocals.
Fourth, tan = sin/cos and cot = cos/sin.
Fifth, when given a ratio but not actual lengths, introduce a variable like x or k, apply Pythagoras theorem, and find all sides proportionally.
Sixth, trigonometrical ratios are pure numbers without units.
And that brings us to the end of this lesson on Trigonometrical Ratios. You have learned the foundation of trigonometry — how angles and sides speak to each other through ratios. Practice identifying these ratios in different triangles, and you will find that what seems complex becomes natural with repetition. Keep your curiosity alive, and I look forward to our next mathematical journey together. Goodbye for now.