Trigonometric Identities Connecting the Angles of a Triangle:
If three or more angles are connected by a relation, then several identities may be obtained involving the trigonometric ratios of the angles. However, if the sum of three given angles is two right angles, then they are the angles of a triangle and in this case, the trigonometric identities that can be established connecting the t-ratios of the angles are more practical importance than the others.
The conclusions which are found to be very important in solving different problems on identities are given below-
(i) If A, B, C are the angles of a triangle, then A + B + C = π ∴ A + B = π – C, B + C = π – A and C + A = π – B ∴ sin (A + B) = sin C, cos (A + B) = -cos C, tan (A + B) = -tan C and so on.
(ii) If A, B, C are the angles of a triangle, then A + B + C = π ⇒ A/2 + B/2 + C/2 = π/2
⇒ (A + B)/2 = π/2 -C/2, (B + C)/2 = π/2 -A/2 and (C + A)/2 = π/2 -B/2
The various types of identities, involving the t-ratios of the angles of a triangle, are discussed below-
Identities involving sines and cosines of multiples or submultiples of the angles:
The working rules for solving these types of problems are-
(i) Any two of the given terms are taken together and expressed as a product of sines and cosines using one of the following formulae-
(ii) The sum of two angles in the product so obtained is expressed in terms of the third angle by using the given condition (i.e. A + B + C = π).
(iii) The third term is then expanded by using one of the formulae. sin 2θ = 2 sin θ cos θ, cos 2θ = 1 -2 sin2 2θ = 2 cos2 θ – 1.
(iv) The common factor from the two terms thus obtained.
(v) The t-ratios of the single angle, in this case, is expressed in terms of the sum of the other two angles by using the given condition.
(vi) One of the C-D formulae is then finally used to change the sum of t-ratios into a product, which gives the final result.
Let us take the following example-
Identities involving squares of sines and cosines of multiples or submultiples of the angles:
In this method, the squares of sines or cosines are changed to cosines of the double angle using the formulae sin2 θ = 1/2 (1 – cos 2θ) and cos2 θ = 1/2 (1 + cos 2θ). The problem then can be solved following the method discussed in type 1.
The following example will be helpful in understanding the method.
Identities involving tangents and cotangents of multiples or submultiples of the angles:
The steps for solving problems of this type are-
(i) The given condition is expressed in such a way that the left-hand side becomes the sum of two multiples or submultiples of the angles appearing in the identity.
(ii) Tangents or cotangents of the angles are taken on both sides.
(iii) Addition formula for tangents or cotangents is used.
(iv) The expressions obtained are multiplied crosswise.
(v) The terms are finally arranged to get the required identity.
Let us consider the identity, ‘If A + B + C = π, then tan A + tan B + tan C = tan A tan B tan C.’
In this case, the given condition is written as A + B = π – C. Then, tangents are taken on both sides-
∴ tan (A + B) = tan (π – C)
⇒ (tan A + tan B)/(1 – tan A tan B) = – tan C (expanding the left hand side)
Multiplying crosswise, we get-
tan A + tan B = – tan C + tan A tan B tan C
Rearranging, we get the required form.
Trigonometrical Identities connecting the angles other than those of triangles:
If the sum of the angles involved in the problem is not equal to two right angles, then also the C-D formulae, the formula for compound angles and other trigonometric relations are used to get the identities.