What is trigonometry?
Trigonometry means "measuring triangles". It links the angles of a right triangle to the lengths of its sides. An acute angle is an angle smaller than 90°. In a right triangle the other two angles are always acute.
Naming the sides: opposite, adjacent, hypotenuse
Take right triangle ABC with the right angle at B. Look from angle A.
- Hypotenuse AC: the longest side, facing the right angle. It never changes.
- Opposite side BC: the side facing angle A.
- Adjacent side AB: the side next to angle A that is not the hypotenuse.
If you look from angle C instead, opposite and adjacent swap places. So always ask: "which angle am I standing at?"
The six trigonometric ratios
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 = AC/BC
- sec A = 1/cos A = AC/AB
- cot A = 1/tan A = AB/BC
Also tan A = sin A / cos A and cot A = cos A / sin A.
Why do the ratios not depend on size? Two right triangles with the same angle A are similar. Similar triangles have their sides in the same proportion, so every ratio stays the same. This is what the 3D shows in step 4.
Since the hypotenuse is the longest side, sin A and cos A are always less than 1 for an acute angle (and sec A, cosec A are more than 1).
Finding all ratios when one is given
If sin A = 3/5, draw a triangle with opposite = 3k and hypotenuse = 5k. By Pythagoras, adjacent = √(25k² − 9k²) = 4k. Now read every ratio: cos A = 4/5, tan A = 3/4, cosec A = 5/3, sec A = 5/4, cot A = 4/3. The k always cancels.
Values at 0°, 30°, 45°, 60°, 90°
| A | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin A | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| cos A | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| tan A | 0 | 1/√3 | 1 | √3 | not defined |
| cosec A | not defined | 2 | √2 | 2/√3 | 1 |
| sec A | 1 | 2/√3 | √2 | 2 | not defined |
| cot A | not defined | √3 | 1 | 1/√3 | 0 |
Where do they come from? 45°: a right triangle with two 45° angles has equal legs, say 1 and 1, so the hypotenuse is √2 and sin 45° = cos 45° = 1/√2. 30° and 60°: cut an equilateral triangle of side 2 in half. You get sides 1, √3 and 2, so sin 30° = 1/2 and sin 60° = √3/2. 0° and 90°: squash the triangle flat (opposite becomes 0) or stand it up (adjacent becomes 0).
Board exams often ask you to evaluate expressions like 2 tan²45° + cos²30° − sin²60° using this table.
Key formulas and definitions
- sin A = opposite / hypotenuse
- cos A = adjacent / hypotenuse
- tan A = opposite / adjacent = sin A / cos A
- cosec A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A
- sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2
- tan 30° = 1/√3, tan 45° = 1, tan 60° = √3
Worked examples
1. In right triangle ABC (right angle at B), AB = 4 cm, BC = 3 cm. Find sin A, cos A and tan A.
Hypotenuse AC = √(4² + 3²) = √25 = 5 cm. From A: opposite = BC = 3, adjacent = AB = 4. sin A = 3/5, cos A = 4/5, tan A = 3/4.
2. If tan A = 5/12, find sin A and cos A.
Opposite = 5k, adjacent = 12k. Hypotenuse = √(25k² + 144k²) = 13k. sin A = 5/13, cos A = 12/13.
3. Evaluate sin 60° cos 30° + sin 30° cos 60°.
(√3/2)(√3/2) + (1/2)(1/2) = 3/4 + 1/4 = 1.
4. Evaluate 2 tan²45° + cos²30° − sin²60°.
2(1)² + (√3/2)² − (√3/2)² = 2 + 3/4 − 3/4 = 2.
5. If sin(A − B) = 1/2 and cos(A + B) = 1/2, with A and B acute and A > B, find A and B.
sin 30° = 1/2, so A − B = 30°. cos 60° = 1/2, so A + B = 60°. Adding: 2A = 90°, A = 45°. Then B = 15°.
6. In triangle PQR, right-angled at Q, PR + QR = 25 cm and PQ = 5 cm. Find sin P, cos P and tan P.
Let QR = x, so PR = 25 − x. Pythagoras: (25 − x)² = 5² + x² → 625 − 50x = 25 → x = 12. So QR = 12, PR = 13. From P: opposite = QR = 12, adjacent = PQ = 5. sin P = 12/13, cos P = 5/13, tan P = 12/5.
7. If 3 cot A = 4, check whether (1 − tan²A)/(1 + tan²A) = cos²A − sin²A.
cot A = 4/3, so tan A = 3/4. Adjacent 4k, opposite 3k, hypotenuse 5k. Left side: (1 − 9/16)/(1 + 9/16) = (7/16)/(25/16) = 7/25. Right side: 16/25 − 9/25 = 7/25. Both are equal, so yes.
Common mistakes
- Mixing up opposite and adjacent. They depend on which angle you are looking from.
- Thinking sin A means sin × A. "sin A" is one symbol: the sine of angle A.
- Writing tan 90° = ∞ or 0. The correct answer is: not defined (you cannot divide by 0).
- Believing a bigger triangle gives bigger sin values. The ratios depend only on the angle.