Trigonometry Basics: Ratios, Identities & Tricks for Class 10
Trigonometry sounds intimidating, but at the Class 10 level it rests on just three ratios and a small table of values. Once you understand how those ratios come from a right-angled triangle and memorise the standard angles, the chapter turns into one of the most reliable sources of marks in your board exam. This guide builds trigonometry from the ground up, gives you a clean memory trick for the angle table, lists the identities you must know, and works through examples.
What is trigonometry?
Trigonometry studies the relationship between the angles and the sides of a triangle. In Class 10 you work only with right-angled triangles. Pick one of the two non-right angles and call it θ. Relative to that angle, the three sides get special names:
- Hypotenuse — the longest side, opposite the 90° angle.
- Opposite (perpendicular) — the side directly across from θ.
- Adjacent (base) — the remaining side next to θ.
The three main trigonometric ratios
The three primary ratios compare pairs of these sides. A famous memory aid is SOH-CAH-TOA:
sin θ = Opposite ÷ Hypotenuse
cos θ = Adjacent ÷ Hypotenuse
tan θ = Opposite ÷ Adjacent
The other three ratios are simply the reciprocals: cosec θ = 1 ÷ sin θ, sec θ = 1 ÷ cos θ, and cot θ = 1 ÷ tan θ. Notice also that tan θ = sin θ ÷ cos θ, which is worth remembering.
The standard-angle table
You must know the values of the ratios for the angles 0°, 30°, 45°, 60° and 90°. These appear in almost every exam.
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | ½ | √3 ÷ 2 | 1 ÷ √3 |
| 45° | 1 ÷ √2 | 1 ÷ √2 | 1 |
| 60° | √3 ÷ 2 | ½ | √3 |
| 90° | 1 | 0 | Not defined |
A trick to rebuild the table
Write the numbers 0, 1, 2, 3, 4 in a row. Divide each by 4 and take the square root: √(0/4), √(1/4), √(2/4), √(3/4), √(4/4). That gives 0, ½, 1/√2, √3/2, 1 — exactly the sin values for 0° to 90°. Reverse the same list for cos. Divide the sin row by the cos row to get tan. With this trick you never have to memorise the table by force.
Key trigonometric identities
Identities are equations that hold true for every value of θ. The three Pythagorean identities are the most important:
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ
All three come from the Pythagoras theorem applied to a right triangle, which is why they share the same shape.
Solved examples
Example 1: Finding ratios from a triangle
In a right triangle the side opposite θ is 3 cm and the hypotenuse is 5 cm. Find sin θ, cos θ and tan θ.
The adjacent side, by Pythagoras, is √(5² − 3²) = √16 = 4 cm. So sin θ = 3/5, cos θ = 4/5, and tan θ = 3/4. This is the familiar 3-4-5 triangle.
Example 2: Using the standard-angle table
Evaluate sin 30° × cos 60° + cos 30° × sin 60°.
Substitute the values: (½ × ½) + (√3/2 × √3/2) = ¼ + ¾ = 1. (This is actually sin 90°, which equals 1 — a neat check.)
Example 3: Applying an identity
If sin θ = 3/5, find cos θ.
Using sin² θ + cos² θ = 1, we get cos² θ = 1 − (3/5)² = 1 − 9/25 = 16/25, so cos θ = 4/5.
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