Trigonometry: Formulas
" Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles. "Trigonometry Ratios ( त्रिकोणमितीय अनुपात )
त्रिकोणमितीय अनुपात एक समकोण त्रिभुज के किनारों के बीच का अनुपात है।
" Trigonometric ratios are the ratios between edges of a right triangle. "
- sinθ = perpendicular (लंम्ब ) / Hypotenuse (कर्ण)
- cosθ = base (आधार) / Hypotenuse (कर्ण)
- tanθ = perpendicular (लंम्ब ) / base (आधार)
- cosecθ = Hypotenuse (कर्ण) / perpendicular (लंम्ब )
- secθ = Hypotenuse (कर्ण) / base (आधार)
- cotθ = base (आधार) / perpendicular (लंम्ब )
Trigonometric Ratio of Certain Angles (कोणों के त्रिकोणमितीय अनुपात)
| Angles ⇒ | 0° | 30° | 45° | 60° | 90° |
| Radians ⇒ | 0 | π/6 | π/4 | π/3 | π/2 |
| Sin θ | 0 | 1/2 | 1/√2 | √3/2 | 1 |
| Cos θ | 1 | √3/2 | 1/√2 | 1/2 | 0 |
| Tan θ | 0 | 1/√3 | 1 | √3 | ∞ |
| Cot θ | ∞ | √3 | 1 | 1/√3 | 0 |
| Sec θ | 1 | 2/√3 | √2 | 2 | ∞ |
| Cosec θ | ∞ | 2 | √2 | 2/√3 | 1 |
Important Formula for Sum and Difference Of Two Angles :
- Sin (A+B) = Sin A Cos B + Cos A Sin B
- Sin (A-B) = Sin A Cos B – Cos A Sin B
- Cos (A+B) = Cos A Cos B – Sin A Sin B
- Cos (A-B) = Cos A Cos B + Sin A Sin B
- 2 sinA.cosB = sin(A+B)+sin (A-B)
- 2 cosA.sinB = sin(A+B)-sin (A-B)
- 2 sinA.sinB = cos(A-B)-cos(A+B)
- 2 cosA.cosB = cos(A+B)+cos(A-B)
Transformation of products and Sum :
Sin A Sin B = ½ [Cos (A-B) – Cos (A+B)]
Cos A Cos B = ½ [Cos (A-B) + Cos (A+B)]
Sin A Cos B = ½ [Sin (A+B) + Sin (A-B)]
Cos A Sin B = ½ [Sin (A+B) – Sin (A-B)]
Sin A + Sin B = 2 sin [(A+B )/2 ] cos [(A−B) /2]
Sin A – Sin B = 2 cos[(A+B) /2] sin [(A−B)/2]
Cos A + Cos B = 2 cos [(A+B )/2 ] cos [(A−B) / 2]
Cos A – Cos B = – 2 sin[(A+B) /2] sin [(A−B) /2]
Different Formula For Tangent :
- tan(A+B) = [(tan A + tan B)/(1 – tan A tan B)]
- tan(A-B) = [(tan A – tan B)/(1 + tan A tan B)]
- cot(A+B) = [(cot A cot B − 1)/(cot B + cot A)]
- cot(A-B) = [(cot A cot B + 1)/(cot B – cot A)]
Sign of Trigonometric Functions in Different Quadrants :
| Quadrants ⇒ | I | II | III | IV |
| Sin A | + | + | – | – |
| Cos A | + | – | – | + |
| Tan A | + | – | + | – |
| Cot A | + | – | + | – |
| Sec A | + | – | – | + |
| Cosec A | + | + | – | – |