There are six fundamental ratios, often known as trigonometry all formula, in the trigonometry study. These ratios can be represented in terms of the six basic trigonometric functions and are defined as the relationship between the lengths of the sides of a right triangle. Sine, cosine, tangent, cotangent, secant, and cosecant are the six functions. Students can use these ratios to solve various trigonometric issues by learning these ratios and the accompanying functions. The six fundamental trigonometric operations-basically offer a framework for examining the angles and sides of a right triangle.
| Angle (radian value) | 0 | π/6 | π/4 | π/3 | π/2 |
| Angle (degree value) | 0० | 30० | 45० | 60० | 90० |
| 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 | undefined (∞) |
| Cot θ | undefined (∞) | √3 | 1 | 1/√3 | 0 |
| Sec θ | 1 | 2/√3 | √2 | 2 | undefined (∞) |
| Cosec θ | undefined (∞) | 2 | √2 | 2/√3 | 1 |
Trigonometric 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
- 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 A + cot B)
- Cot (A – B) = (cot A . cot B + 1) / (cot B – cot A)
Formula for sum of two trigonometric functions
Formula for the product of two trigonometric functions
- 2 sin A . cos B = sin (A + B) + sin (A – B)
- 2 cos A . sin B = sin (A + B) – sin (A – B)
- 2 sin A . sin B = cos (A – B) – cos (A + B)
- 2 cos A . cos B = cos (A + B) + cos (A – B)
Formula For Two Trigonometric Angles
- Sin 2α = 2 sin α . cos α = 2 tan α / (1 + tan2 α)
- cos 2α = (cos2 α – sin2 α)= (2 cos2 α – 1) = (1 – 2 sin2 α)
- cos 2α = [(1- tan2 α) / (1 + tan2 α)]
- tan 2α = [(2 tan α) / (1- tan2 α) ]
Formula Of Three Trigonometric Angles
- Sin 3Α = 3 sin A – 4 sin 3 A
- cos 3Α = 4 cos 3 A – 3 cos A
Identities of Trigonometry Half Angles
- Sin β = 2 sin(β/2) . cos(β/2)
- Cos β = [cos2 (β/2)- sin2 (β/2)]= [2 cos2(β/2) – 1] = [1 – 2 sin2 (β/2)]
- cos β = [{1- tan2 (β/2)}/ {1 + tan2 (β/2)}]
- tan β = [{2 tan (β/2)} / {1- tan2 (β/2)} ]
Trigonometric identities in all four quadrants
Values of Trigonometric Ratios in First Quadrant
| (900 – θ) values of functions for | (3600 + θ) values of functions for |
| Sin (900 – θ) = Cos θ
Cos (900 – θ) = Sin θ Tan (900 – θ) = Cot θ Sec (900 – θ) = Cosec θ Cot (900 – θ) = Tan θ Cosec(900-θ)= Sec θ |
Sin (3600 + θ) = Sin θ
Cos (3600 + θ) = Cos θ Tan (3600 + θ) = Tan θ Sec (3600 + θ) = Sec θ Cot (3600 + θ) = Cot θ Cosec (3600+θ) = Cosec θ |
Values of Trigonometric Ratios in Second Quadrant
Sin ↔ cos and Cosec ↔ Sec for positive (900 + θ)
Sin ↔ Sin and Cosec ↔ Cosec for positive (1800 – θ)
| (900 + θ) values of functions for | (1800 – θ) values of functions for |
| Sin (900 + θ) = Cos θ
Cos (900 + θ) = – Sin θ Tan (900 + θ) = – Cot θ Sec (900 + θ) = – Cosec θ Cot (900 + θ) = – Tan θ Cosec (900+θ) = Sec θ |
Sin (1800 – θ) = Sin θ
Cos (1800 – θ) = – Cos θ Tan (1800 – θ) = – Tan θ Sec (1800 – θ) = – Sec θ Cot (1800 – θ) = – Cot θ Cosec (1800-θ)= Cosec θ |
Value of trigonometric ratios in the third quadrant :-
Tan ↔ Tan and Cot ↔ Cot for positive (1800 + θ)
Tan ↔ Cot and Cot ↔ Tan for positive (2700 – θ)
| (1800 + θ) values of functions for | (2700 – θ) values of functions for |
| Sin (1800 + θ) = – Sin θ
Cos (1800 + θ) = – Cos θ Tan (1800 + θ) = + Tan θ Sec (1800 + θ) = – Sec θ Cot (1800 + θ) = + Cot θ Cosec (1800+θ) = -Cosec θ |
Sin (2700 – θ) = – Cos θ
Cos (2700 – θ) = – Sin θ Tan (2700 – θ) = + Cot θ Sec (2700 – θ) = – Cosec θ Cot (2700 – θ) = + Tan θ Cosec (2700-θ)= -Sec θ |
Values of Trigonometric Ratios in Quadrants
Cos ↔ Sin and Sec ↔ Cosec for positive (2700 + θ)
Cos ↔ Cos and Sec ↔ Sec for positive (3600 – θ)
| (2700 + θ) values of functions for | (3600 – θ) values of functions for |
| Sin (2700 + θ) = – Cos θ
Cos (2700 + θ) = + Sin θ Tan (2700 + θ) = – Cot θ Sec (2700 + θ) = + Cosec θ Cot (2700 + θ) = – Tan θ Cosec (2700+θ) = – Sec θ |
Sin (3600 – θ) = – Sin θ
Cos (3600 – θ) = + Cos θ Tan (3600 – θ) = – Tan θ Sec (3600 – θ) = + Sec θ Cot (3600 – θ) = – Cot θ Cosec (3600-θ)= – Cosec θ |
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