Trigonometry formulas for Class 10
Class 10 trigonometry rests on three things: the six ratios, the table of standard values, and three identities. Learn these by heart and the proofs and height-and-distance questions become routine.
Short notes
- In a right triangle, for angle A: sin A = perpendicular / hypotenuse, cos A = base / hypotenuse, tan A = perpendicular / base.
- The other three are reciprocals: cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A.
- tan A = sin A / cos A, and cot A = cos A / sin A.
- Identity 1: sin²A + cos²A = 1.
- Identity 2: 1 + tan²A = sec²A (for A < 90°).
- Identity 3: 1 + cot²A = cosec²A (for 0° < A ≤ 90°).
- The value of sin A or cos A is never more than 1; sec A and cosec A are never less than 1.
- As A goes from 0° to 90°, sin A increases from 0 to 1 and cos A decreases from 1 to 0.
- tan 90° and cot 0° are not defined, nor are sec 90° and cosec 0°.
- Angle of elevation: measured up from the horizontal, when you look up at an object. Angle of depression: measured down from the horizontal, when you look down.
- In height-and-distance questions, draw the right triangle first, then use tan (opposite/adjacent) when the height and the distance on the ground are involved.
At a glance
| 0° | 30° | 45° | 60° | 90° | |
|---|---|---|---|---|---|
| sin | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan | 0 | 1/√3 | 1 | √3 | Not defined |
| cosec | Not defined | 2 | √2 | 2/√3 | 1 |
| sec | 1 | 2/√3 | √2 | 2 | Not defined |
| cot | Not defined | √3 | 1 | 1/√3 | 0 |
Memory tricks
The ratios: "Pandit Badri Prasad, Har Har Bole"
Read the top words, then the bottom: sin = P/H, cos = B/H, tan = P/B.
The sin row in 5 seconds
Write √0, √1, √2, √3, √4 and divide each by 2: 0, 1/2, 1/√2, √3/2, 1. The cos row is the same, backwards.
tan from sin and cos
tan = sin ÷ cos for each column: at 60°, (√3/2) ÷ (1/2) = √3.
Practice questions
The value of sin 30° + cos 60° is:
Show answer
1. 1/2 + 1/2 = 1.
If sin A = 3/5, then cos A is:
Show answer
4/5. cos²A = 1 − 9/25 = 16/25, so cos A = 4/5.
sec²A − tan²A equals:
Show answer
1. From 1 + tan²A = sec²A.
A tower casts a shadow equal to its height. The sun's angle of elevation is:
Show answer
45°. tan θ = height / shadow = 1, so θ = 45°.
Questions
What is the quickest way to learn the trigonometry table?
Learn only the sin row with the √0 to √4 over 2 trick. Reverse it for cos, divide sin by cos for tan, and flip each for cosec, sec and cot.
Which identities are used in Class 10 proofs?
Just three: sin²A + cos²A = 1, 1 + tan²A = sec²A and 1 + cot²A = cosec²A. Most proofs turn everything into sin and cos first.
Can TellBrief explain a trigonometry question from my book?
Yes. Add a photo of the page, choose CBSE once, and ask about any step in Hindi or English. The answer comes from your page.