All Basic Trigonometric Identities Formulas -
We already know all basic formulas of trigonometry (Trigonometric Identities maths trigonometry), such as Pythagorean trigonometry identities, Reciprocal & Quotient Identities for trigonometric functions, Co-function identities (shifting angles), Even and Odd Angle Formulas, Trigonometry table formulas for angles, Trigonometrical ratios of Angle (90°+θ) in terms of those of θ for all values of θ, etc.
In this article, we learn all basic Trigonometric Identities Formulas in Maths: such as Sum and Difference identities, Double Angle identities, Half-Angle identities, Thrice of Angle Formulas, Product to Sum formulas, Sum to Product formulas etc. are given.
Students in Classes 10, 11, and 12 will benefit from learning and memorising these trigonometry math formulas in order to get success in this topic.
Trigonometric Identities in Maths:
Trigonometric identities in Maths are equations that contain trigonometric functions that hold true for every value of the variables involved.
1. Pythagorean Identities in Trigonometry:
The Pythagorean Theorem is the source of some of the most commonly used trigonometric identities, such as:
| Name of Identity | Formula |
|---|---|
| First (Basic) Identity - (Primary Pythagorean Identity) |
sin²θ + cos²θ = 1 |
| Second Identity - (Tangent–Secant Identity) (Dividing First Identity by cos²θ) |
1 + tan²θ = sec²θ |
| Third Identity - (Cotangent–Cosecant Identity) (Dividing First Identity by sin²θ) |
1 + cot²θ = csc²θ |
These are called Pythagorean Identities in trigonometry because they are derived from the relation of the sides of a right-angled triangle using the Pythagoras theorem.
The Bhaskaracharya Sum and Difference Identities in Trigonometry:
These identities help us find the sine, cosine, tangent, or cotangent of the sum or difference of two angles.
1. Sum & Difference Identities for Sin & Cos functions in Trigonometry
Sum and Difference Identities for Sine and Cosine functions in trigonometry for any two angles A and B are as follows:
1. \(\sin(A+B) = \sin A \cos B + \cos A \sin B\)
2. \(\sin(A-B) = \sin A \cos B - \cos A \sin B\)
3. \(\cos(A+B) = \cos A \cos B - \sin A \sin B\)
4. \(\cos(A-B) = \cos A \cos B + \sin A \sin B\)
2. Sum and Difference Identities for Tan and Cot functions in trigonometry
Sum and Difference Identities for Tan and Cot functions in trigonometry for any two angles A and B are as follows:
1. \(\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\)
2. \(\tan(A-B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\)
3. \(\cot(A+B) = \frac{\cot A \cot B - 1}{\cot A + \cot B}\)
4. \(\cot(A-B) = \frac{\cot A \cot B + 1}{\cot B - \cot A}\)
3. Sum Identities for three angles A, B and C
When we have A + B + C, we expand step by step using the two-angle formulas.
Sum Identities for three angles A, B and C for Sine, Cosine and Tan functions in trigonometry are as follows:
1. \(\sin(A+B+C)\)
2. \(\cos(A+B+C) \)
3. \(\tan(A+B+C) \)
\(= \frac{\tan A + \tan B + \tan C - \tan A \tan B \tan C}{1 - \tan A \tan B - \tan B \tan C - \tan C \tan A}\)
Notice: The numerator is “sum of tangents – product of all three”, and the denominator is “1 – sum of pairwise products”.
Double Angle Identities in Trigonometry:
The double angle identities are just special cases of Bhaskaracharya's Sum and Difference formulas, when A = B.
A trigonometric expression can be written in terms of a single trigonometric function using the double angle identities. Here, trigonometrical ratios of angle 2A in terms of those of angle A for all values of A are given.
Double Angle Identities for Sine and Cosine functions in trigonometry are as follows:
1. \(\sin(2A) = 2 \sin A \cos A\)
2 (i). \(\cos(2A) = \cos^2 A - \sin^2 A\)
(ii). \(\cos(2A) = 2\cos^2 A - 1\)
(iii). \(\cos(2A) = 1 - 2\sin^2 A\)
3. \(\tan(2A) = \frac{2\tan A}{1 - \tan^2 A}\)
4. \(\cot(2A) = \frac{\cot^2 A - 1}{2\cot A}\)
5. \(\sec(2A) = \frac{1}{\cos(2A)}\)
6. \(\csc(2A) = \frac{1}{\sin(2A)}\)
Half-Angle Identities in Trigonometry:
The half-angle identities are also special cases of Bhaskaracharya's Sum and Difference Formulas.
Half-angle identities can be used to evaluate the trigonometrical function of an angle that is not on the unit circle. For example, we can find the value of the trigonometrical function of 15°, which is not on the unit circle, because 15° is half of 30°, which is on the unit circle.
Here, trigonometrical ratios of angle A in terms of those of angle A/2 for all values of A are given.
1. \(\sin\frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{2}}\)
2. \(\cos\frac{A}{2} = \pm\sqrt{\frac{1 + \cos A}{2}}\)
3 (i). \(\tan\frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{1 + \cos A}}\)
(ii). \(\tan\frac{A}{2} = \frac{\sin A}{1 + \cos A}\)
(iii). \(\tan\frac{A}{2} = \frac{1 - \cos A}{\sin A}\)
Thrice of Angle Identities in Trigonometry (Triple Angle Identities):
The thrice of angle identities, that is, trigonometrical ratios of angle 3A in terms of those of angle A for all values of A are as follows:
1. \(\sin(3A) = 3\sin A - 4\sin^3 A\)
2. \(\cos(3A) = 4\cos^3 A - 3\cos A\)
3. \(\tan(3A) = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A}\)
4. \(\cot(3A) = \frac{\cot^3 A - 3\cot A}{3\cot^2 A - 1}\)
Sum to Product Identities in Trigonometry
These identities convert sums into products, which often simplify trigonometric equations.
Sum to Product formulas or identities for two angles C and D in Trigonometry are as follows:
2. \(\sin C - \sin D = 2 \cos\left(\frac{C+D}{2}\right) \sin\left(\frac{C-D}{2}\right)\)
3. \(\cos C + \cos D = 2 \cos\left(\frac{C+D}{2}\right) \cos\left(\frac{C-D}{2}\right)\)
4. \(\cos C - \cos D = -2 \sin\left(\frac{C+D}{2}\right) \sin\left(\frac{C-D}{2}\right)\)
Product to Sum Identities
In summary:
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