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Trigonometry is new in Class 10, and for many students it is the first chapter where a single angle stands for a whole family of numbers. It becomes manageable once you see that every ratio is just one side of a right triangle divided by another, and that the standard values come from two triangles you can draw in a minute. This post covers the current syllabus, how to build the values table, worked examples and the slips that cost marks.
What does the syllabus cover?
CBSE's Class X Mathematics syllabus for 2026-27 lists trigonometric ratios of an acute angle of a right triangle, the ratios "whichever are defined at 0° and 90°", the values at 30°, 45° and 60°, relationships between the ratios, and the identity sin²A + cos²A = 1 with "only simple identities to be given". Trigonometry, including heights and distances, carries 12 of the 80 theory marks. The NCERT chapter, rationalised and reprinted for 2026-27, has sections on the ratios, specific angles and identities. Trigonometric ratios of complementary angles are not in the current chapter.
What are the six ratios?
In a right triangle ABC with the right angle at B, for angle A:
sin A = opposite/hypotenuse = BC/AC
cos A = adjacent/hypotenuse = AB/AC
tan A = opposite/adjacent = BC/AB
cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A
tan A = sin A / cos A
The ratios depend only on the angle, not the size of the triangle, because triangles with the same angles are similar.
Example 1. If sin A = 3/5, find the other five ratios.
Take opposite = 3k and hypotenuse = 5k.
Adjacent = √(25k² − 9k²) = √(16k²) = 4k.
cos A = 4/5, tan A = 3/4, cosec A = 5/3, sec A = 5/4, cot A = 4/3.
Check: sin²A + cos²A = 9/25 + 16/25 = 1.
How do you build the values table without memorising it?
Draw two triangles.
For 45°: a right isosceles triangle with legs 1 and 1. The hypotenuse is √2. So sin 45° = cos 45° = 1/√2 and tan 45° = 1.
For 30° and 60°: half of an equilateral triangle with side 2. The sides are 1, √3 and 2. So sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3, and sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3.
At 0° and 90°: sin 0° = 0, cos 0° = 1, sin 90° = 1, cos 90° = 0. tan 90° is not defined, because cos 90° = 0 and you cannot divide by zero.
Worked examples with the values
Example 2. Evaluate 2 tan²45° + cos²30° − sin²60°.
tan 45° = 1, so 2 tan²45° = 2.
cos²30° = (√3/2)² = 3/4. sin²60° = (√3/2)² = 3/4.
Total: 2 + 3/4 − 3/4 = 2.
Example 3. Prove that (1 − cos²A)(1 + cot²A) = 1.
1 − cos²A = sin²A, from sin²A + cos²A = 1.
1 + cot²A = cosec²A. (Divide sin²A + cos²A = 1 by sin²A.)
So the left side is sin²A × cosec²A = sin²A × 1/sin²A = 1.
Example 4. Prove that (sec A − tan A)(sec A + tan A) = 1.
The left side is a difference of squares: sec²A − tan²A.
Divide sin²A + cos²A = 1 by cos²A to get tan²A + 1 = sec²A, so sec²A − tan²A = 1.
In identity proofs, work on one side only and end with "= right-hand side". Do not move terms across the equals sign as if solving an equation.
Which mistakes lose marks?
Treating sin A as sin times A. sin A is one number. You cannot cancel the A in sin A / A.
Splitting sin(A + B). sin(30° + 30°) = sin 60° ≈ 0.866, but sin 30° + sin 30° = 1. They are not equal.
sin²A versus sin A². sin²A means (sin A)².
Confusing opposite and adjacent. Mark the angle first, then label sides from it.
Using tan 90°. It is not defined. Say so.
Working on both sides of an identity at once. Start from the more complicated side.
If an identity will not come out, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning, so you can see which identity was used at each step and why.
How should you revise the values and identities?
Self-testing beats rereading for anything you must recall in an exam. Roediger and Karpicke (2006) had students either restudy prose passages or take recall tests on them. After five minutes, restudying looked better. On tests two days and one week later, the students who had tested themselves remembered much more, even though the restudy group felt more confident.
A one-week routine:
Day 1: Draw the two triangles and build the full table from them. Do not copy it from the book.
Day 2: Rebuild the table from a blank page in under three minutes. Check it.
Day 3: Five "given one ratio, find the rest" questions, each checked with sin²A + cos²A = 1.
Day 4: Five evaluation questions using the table, then rebuild the table again.
Day 5: Five simple identity proofs, writing which identity you used on each line.
For more identity practice, read trigonometric identities practice. The chapter that uses all this is heights and distances in Class 10.
Today, close your book and build the values table from the two triangles on a blank page.
