CHAPTER
Trigonometric Ratios
4 and Identities
1. Trigonometric Ratio – Basic Terminology 2. Trigonometric identities:
(i) sin2θ + cos2θ = 1
(ii) sec2θ – tan2θ = 1
hypotenuse (iii) cosec2θ – cot2θ = 1
opposite
3. Allied angles:
Two angles are said to be allied when their sum or difference
q π
is either zero or a multiple of , two angles x, y are allied
nπ 2
adjacent C
angles iff =x ± y 0 or ,n ∈N .
The six trigonometric ratios of θ are defined as follows: 2
opposite adjacent opposite
= sin θ = , cos θ = , tan θ
hypotenuse hypotenuse adjacent
adjacent hypotenuse hypotenus
= cot θ = , cosecθ = , sec θ
opposite opposite adjacent
π π 3π 3π
q→ −θ +θ p–q p+q −θ +θ 2p – q 2p + q –q
2 2 2 2
sin cos q cos q sin q –sin q –cos q –cos q –sin q sin q –sin q
cos sin q –sin q –cos q –cos q –sin q sin q cos q cos q cos q
tan cot q –cot q – tan q tan q cot q –cot q –tan q tan q –tan q
cot tan q –tan q – cot q cot q tan q –tan q –cot q cot q –cot q
sec cosecq –cosecq –sec q –sec q –cosec q cosecq sec q sec q sec q
cosec sec q sec q cosec q –cosec q –sec q –sec q –cosec q cosec q –cosec q
4. Sum & Difference Formula 5. Product to sum
(i) sin (A + B) = sin A cos B + cos A sin B (i) 2 sin A cos B = sin (A + B) + sin (A – B)
(ii) sin (A – B) = sin A cos B – cos A sin B
(iii) cos (A + B) = cos A cos B – sin A sin B (ii) 2 cos A sin B = sin (A + B) – sin (A – B)
(iv) cos (A – B) = cos A cos B + sin A sin B (iii) 2 cos A cos B = cos (A + B) + cos (A – B)
tan A + tan B (iv) 2 sin A sin B = cos (A – B) – cos (A + B)
(v) tan( A + B) =
1 − tan A tan B 6. Trigonometric transformations:
tan A − tan B C + D C − D
(vi) tan( A − B) = (i) sin C + sin D =2sin cos
1 + tan A tan B 2 2
cot B cot A − 1 C + D C − D
(vii) cot( A + B) = (ii) sin C − sin D =
2 cos sin
cot B + cot A 2 2
cot B cot A + 1 C + D C − D
(viii) cot( A − B ) = (iii) cos C + cos D =
2 cos cos
cot B − cot A 2 2
Trigonometric Ratios
4 and Identities
1. Trigonometric Ratio – Basic Terminology 2. Trigonometric identities:
(i) sin2θ + cos2θ = 1
(ii) sec2θ – tan2θ = 1
hypotenuse (iii) cosec2θ – cot2θ = 1
opposite
3. Allied angles:
Two angles are said to be allied when their sum or difference
q π
is either zero or a multiple of , two angles x, y are allied
nπ 2
adjacent C
angles iff =x ± y 0 or ,n ∈N .
The six trigonometric ratios of θ are defined as follows: 2
opposite adjacent opposite
= sin θ = , cos θ = , tan θ
hypotenuse hypotenuse adjacent
adjacent hypotenuse hypotenus
= cot θ = , cosecθ = , sec θ
opposite opposite adjacent
π π 3π 3π
q→ −θ +θ p–q p+q −θ +θ 2p – q 2p + q –q
2 2 2 2
sin cos q cos q sin q –sin q –cos q –cos q –sin q sin q –sin q
cos sin q –sin q –cos q –cos q –sin q sin q cos q cos q cos q
tan cot q –cot q – tan q tan q cot q –cot q –tan q tan q –tan q
cot tan q –tan q – cot q cot q tan q –tan q –cot q cot q –cot q
sec cosecq –cosecq –sec q –sec q –cosec q cosecq sec q sec q sec q
cosec sec q sec q cosec q –cosec q –sec q –sec q –cosec q cosec q –cosec q
4. Sum & Difference Formula 5. Product to sum
(i) sin (A + B) = sin A cos B + cos A sin B (i) 2 sin A cos B = sin (A + B) + sin (A – B)
(ii) sin (A – B) = sin A cos B – cos A sin B
(iii) cos (A + B) = cos A cos B – sin A sin B (ii) 2 cos A sin B = sin (A + B) – sin (A – B)
(iv) cos (A – B) = cos A cos B + sin A sin B (iii) 2 cos A cos B = cos (A + B) + cos (A – B)
tan A + tan B (iv) 2 sin A sin B = cos (A – B) – cos (A + B)
(v) tan( A + B) =
1 − tan A tan B 6. Trigonometric transformations:
tan A − tan B C + D C − D
(vi) tan( A − B) = (i) sin C + sin D =2sin cos
1 + tan A tan B 2 2
cot B cot A − 1 C + D C − D
(vii) cot( A + B) = (ii) sin C − sin D =
2 cos sin
cot B + cot A 2 2
cot B cot A + 1 C + D C − D
(viii) cot( A − B ) = (iii) cos C + cos D =
2 cos cos
cot B − cot A 2 2
