Trigonometric Ratios (Class 10): sin, cos, tan Explained

In one line: the trigonometric ratios are six fixed ratios of the sides of a right-angled triangle — sin, cos, tan, cosec, sec and cot — measured with respect to one of its acute angles.

For an acute angle A in a right triangle, the three primary ratios are sin A = opposite ÷ hypotenuse, cos A = adjacent ÷ hypotenuse, and tan A = opposite ÷ adjacent. The other three — cosec A, sec A and cot A — are simply the reciprocals of these. That is the entire idea: pick an angle, look at how the three sides relate to it, and you have all six ratios.

The single most important fact is that these ratios depend only on the angle, not on the size of the triangle. A 30° angle gives the same sin, cos and tan whether the triangle is tiny or huge, because enlarging a right triangle keeps its sides in the same proportion. That is what makes trigonometry powerful — measure one angle and one side, and every other length follows. This guide defines all six ratios, gives you the SOH-CAH-TOA memory trick, and shows how to find every ratio from just one. If you want the full chapter map first, see the Introduction to Trigonometry (Class 10) guide.

What are the trigonometric ratios?

A trigonometric ratio is a ratio between two sides of a right-angled triangle, taken with respect to one of its acute angles. In a right triangle, the three sides have names that depend on which acute angle you are looking at:

• Hypotenuse — the longest side, always opposite the right angle. This never changes.
• Opposite side — the side directly across from the angle you have chosen.
• Adjacent side — the side next to the chosen angle (the one that is not the hypotenuse).

Switch your attention to the other acute angle and the opposite and adjacent sides swap roles, while the hypotenuse stays put. Getting these three names right for the angle in question is the whole battle — every ratio is built from them.

🎬 Watch — 30-sec Short: Can you identify the side opposite angle A? 🤔

The three primary ratios: sin, cos, tan

For an acute angle A, the three primary trigonometric ratios are defined as follows. Each is given with its meaning, its variables, and the condition under which it applies, plus a worked value.

• sin A = opposite ÷ hypotenuse. The sine of the angle. Condition: A is an acute angle of a right triangle. In a 3-4-5 triangle where the side opposite A is 3 and the hypotenuse is 5, sin A = 3/5 = 0.6.
• cos A = adjacent ÷ hypotenuse. The cosine of the angle. Condition: same right triangle. With the adjacent side 4 and hypotenuse 5, cos A = 4/5 = 0.8.
• tan A = opposite ÷ adjacent. The tangent of the angle. Condition: the adjacent side is non-zero. With opposite 3 and adjacent 4, tan A = 3/4 = 0.75.

A useful relationship falls straight out of these: tan A = sin A ÷ cos A, because (opposite/hypotenuse) ÷ (adjacent/hypotenuse) = opposite/adjacent. Check it on the numbers above: (3/5) ÷ (4/5) = 3/4. ✓

SOH-CAH-TOA: the memory trick

The fastest way to lock in the three primary ratios is the code word SOH-CAH-TOA:

• SOH → Sin = Opposite / Hypotenuse
• CAH → Cos = Adjacent / Hypotenuse
• TOA → Tan = Opposite / Adjacent

Say it out loud a few times and it sticks for the whole exam. Whenever a problem gives you two sides and an angle, SOH-CAH-TOA tells you instantly which ratio connects them. Practising this reflex until it is automatic is exactly the kind of quick, repeatable drill you can run on examfront’s Topic Practice so it becomes second nature before the exam.

🎬 Watch — 30-sec Short: Can you identify the trig ratio from Opposite ÷ Hypotenuse? 🤔

The three reciprocal ratios: cosec, sec, cot

The remaining three ratios are just the reciprocals (the “flip”) of the first three:

• cosec A = 1 ÷ sin A = hypotenuse ÷ opposite. Read as “cosecant.” With sin A = 3/5, cosec A = 5/3.
• sec A = 1 ÷ cos A = hypotenuse ÷ adjacent. Read as “secant.” With cos A = 4/5, sec A = 5/4.
• cot A = 1 ÷ tan A = adjacent ÷ opposite. Read as “cotangent.” With tan A = 3/4, cot A = 4/3. Equivalently, cot A = cos A ÷ sin A.

A common exam trap is the pairing: notice that sec goes with cos and cosec goes with sin — the “co” swaps sides. A memory line that helps is “sec-cos, cosec-sin — the co crosses over.”

Exam tip: sin A means “the sine of angle A.” It is not sin × A. Writing “sin” on its own, separated from an angle, has no meaning — always keep the ratio attached to its angle.

Why the ratios don’t depend on the triangle’s size

Here is the property that makes trigonometry work: for a fixed angle, the value of each ratio is the same no matter how big or small the triangle is.

The reason is similarity. If you take a right triangle and draw a smaller (or larger) right triangle inside it that shares the same acute angle, the two triangles are similar — their corresponding sides are in the same proportion. Since each trigonometric ratio is a ratio of two sides, and both sides scale by the same factor, the ratio itself is unchanged. So sin 30° is always 1/2, whether the triangle spans a centimetre or a kilometre. Understanding why this holds is what separates memorising from mastering, and it connects directly to the similarity results you met in Triangles (Class 10).

Worked example 1 — find all ratios from one

Problem: In right triangle ABC (right-angled at B), sin A = 3/5. Find the other five trigonometric ratios of A.

• Step 1 — label two sides. sin A = opposite/hypotenuse = 3/5, so let the opposite side BC = 3k and hypotenuse AC = 5k (k is any positive number).
• Step 2 — find the third side. By the Pythagoras theorem, AB² = AC² − BC² = (5k)² − (3k)² = 25k² − 9k² = 16k², so the adjacent side AB = 4k.
• Step 3 — read off the ratios. cos A = 4k/5k = 4/5; tan A = 3k/4k = 3/4; cosec A = 5/3; sec A = 5/4; cot A = 4/3.

Every ratio came from one given value plus Pythagoras. This “one ratio unlocks all six” method is one of the most common 2–3 mark questions in the chapter.

🎬 Watch — 30-sec Short: Can you find cos A from sin A = 3/5? 🤔

Worked example 2 — starting from tan

Problem: Given tan A = 4/3, find sin A and cos A.

• Step 1: tan A = opposite/adjacent = 4/3, so opposite BC = 4k and adjacent AB = 3k.
• Step 2: Hypotenuse AC² = (4k)² + (3k)² = 16k² + 9k² = 25k², so AC = 5k.
• Step 3: sin A = 4k/5k = 4/5 and cos A = 3k/5k = 3/5.

Notice it is the same 3-4-5 triangle as Example 1, just entered through a different door. Whichever ratio you are handed, the recipe is identical: label two sides, use Pythagoras for the third, then read the rest.

Can a ratio be bigger than 1?

Because the hypotenuse is the longest side of a right triangle, the opposite and adjacent sides are always shorter than it. So:

• sin A ≤ 1 and cos A ≤ 1 always — they can never exceed 1.
• sec A ≥ 1 and cosec A ≥ 1 always — being reciprocals of numbers that are at most 1.
• tan A and cot A have no such limit — they can be any positive value.

This is a quick sanity check in the exam: if you ever calculate sin A = 5/3, you have made a slip, because no sine can be greater than 1. Spotting impossible values instantly is exactly the kind of self-check examfront’s Mistake Identification trains you to make automatically.

Going further

This guide covers the core of trigonometric ratios: the six definitions, SOH-CAH-TOA, the reciprocal and quotient links, the size-independence property, and the “find all ratios from one” method. The harder variations — proving that two angles are equal from equal ratios (like sin B = sin Q ⟹ ∠B = ∠Q), multi-step problems where a side is given as a sum or difference (such as OQ − PQ = 1), and mixed questions that combine several ratios in one expression — are where extra guided practice really pays off. Work through the full graded set with step-by-step solutions and personalised help inside examfront, where Chapter Quizzes and Progress Tracking show you exactly which ratio types still trip you up. Next, lock in the identities that connect these ratios in Trigonometric Identities (Class 10).

Key Takeaways

• The six trigonometric ratios — sin, cos, tan, cosec, sec, cot — are ratios of the sides of a right triangle taken with respect to an acute angle.
• SOH-CAH-TOA: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.
• cosec, sec and cot are the reciprocals of sin, cos and tan; also tan = sin/cos and cot = cos/sin.
• A ratio’s value depends only on the angle, not the triangle’s size (because of similar triangles).
• Given one ratio, label two sides, use Pythagoras for the third, and read off all the others.
• sin and cos are always ≤ 1; sec and cosec are always ≥ 1.

Quick Facts

• Trigonometric ratio: a ratio of two sides of a right triangle for a given acute angle.
• sin A = opposite/hypotenuse; cos A = adjacent/hypotenuse; tan A = opposite/adjacent.
• cosec A = 1/sin A; sec A = 1/cos A; cot A = 1/tan A.
• tan A = sin A/cos A; cot A = cos A/sin A.
• Hypotenuse: longest side, opposite the right angle.
• sin A ≤ 1, cos A ≤ 1; sec A ≥ 1, cosec A ≥ 1.
• Chapter: Introduction to Trigonometry · Class: 10 · Subject: Maths · Board: CBSE.

Common Mistakes

1. Mixing up opposite and adjacent. Students label the sides before checking which angle they are working with. Always identify the angle first, then “opposite” is across from it and “adjacent” is beside it.
2. Treating sin A as sin × A. sin A is a single quantity — the sine of A. It cannot be split; “sin” without an angle is meaningless.
3. Flipping the wrong reciprocal. Pairing sec with sin or cosec with cos. Remember: sec ↔ cos, cosec ↔ sin, cot ↔ tan.
4. Forgetting Pythagoras for the third side. When one ratio is given, students try to guess the missing side. Use AC² = AB² + BC² every time.
5. Accepting sin A > 1. Writing an answer like sin A = 5/4. Since the hypotenuse is longest, sin and cos can never exceed 1 — recheck the calculation.

FAQ

Q. What are the trigonometric ratios in Class 10? They are six ratios of the sides of a right triangle with respect to an acute angle: sin, cos, tan, cosec, sec and cot. For an angle A, sin A = opposite/hypotenuse, cos A = adjacent/hypotenuse, tan A = opposite/adjacent, and cosec, sec, cot are their reciprocals.

Q. What is the easiest way to remember sin, cos and tan? Use SOH-CAH-TOA: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. “Opposite” is across from the angle, “adjacent” is next to it, and the hypotenuse is the longest side.

Q. What are the reciprocals of sin, cos and tan? cosec A = 1/sin A, sec A = 1/cos A, and cot A = 1/tan A. In addition, tan A = sin A/cos A and cot A = cos A/sin A.

Q. If one trigonometric ratio is given, can you find the others? Yes. Label the two sides the given ratio provides (using a variable k), use the Pythagoras theorem to find the third side, then read off the remaining ratios. For example, from sin A = 3/5 you get a 3-4-5 triangle and every other ratio.

Q. Can sin A or cos A be greater than 1? No. The hypotenuse is the longest side, so opposite/hypotenuse and adjacent/hypotenuse are always ≤ 1. But sec A and cosec A are always ≥ 1, and tan A can be any positive value.

Related Concepts

• Trigonometric Identities (Class 10): the identities (like sin²A + cos²A = 1) that link the ratios defined here.
• Introduction to Trigonometry (Class 10): the full Chapter 8 roadmap, including standard-angle values.
• Triangles (Class 10): the similarity results that explain why ratios don’t depend on size.
• Heights and Distances (Class 10): where these ratios are used to find real heights and distances.

Ready to master the trigonometric ratios? Drill sin, cos and tan on examfront’s Topic Practice, and let the Practice Companion, Chapter Quizzes and Progress Tracking carry you from the basics to exam-level questions. Start on examfront →