Trigonometric Ratios and Identities
Trigonometric Ratios
Definitions
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The trigonometric ratios of angle θ are
Formulae
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Reciprocal Trigonometric Ratios:
- csc θ = or
- sec θ = or
- cot θ = or
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Trigonometric-ratios of allied (or related) angles
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T-ratios of other allied angles in terms of theta:
Trigonometric Identities
Formulae
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Reciprocal trigonometric identities:
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Pythagorean trigonometric identities:
- sin2 θ + cos2 θ = 1
- sin2 θ = 1 – cos2 θ
- cos2 θ = 1 – sin2 θ
- 1 + tan2 θ = sec2 θ
- sec2 θ – tan2 θ = 1
- 1 + cot2 θ = cosec2 θ
- cosec2 θ – cot2 θ = 1
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Quotient Identities:
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Trigonometric Ratios of Compound 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
- sin(A + B) . sin(A – B) = sin2 A – sin2 B = cos2 B – cos 2 A
- cos(A + B) . cos(A – B) = cos2 A – sin2 B = cos2 B – sin2 A
- sin(A + B + C) = sin A . cos B . cos C + cos A . sin B . cos C + cos A . cos B . sin C – sin A . sin B . sin C
- cos(A + B + C) = cos A . cos B . cos C – cos A . sin B . sin C – sin A . cos B . sin C – sin A . sin B . cos C
- Values of other specific trigonometric angles:
- sin 15° = cos 75° =
- cos 15° = sin 75° =
- tan 15° = cot 75° = 2 – √3
- cot 15° = tan 75° = 2 + √3
- sec 15° = cosec 75° = √6 – √2
- cosec 15° = sec 75° = √6 + √2
- tan(7°) = √(2) – √(3) – √(4) + √(6) = cot(82°)
- cot(7°) = √(2) + √(3) + √(4) + √(6) = tan(82°)
- 4cos36° + cot7° = 1 + √(2) + √(3) + √(4) + √(5) + √(6)
- sin9° = = cos 81°
- cos9° = = sin 81°
- Trigonometric ratios of multiple & sub-multiple angles:
sin A = 2 sin cos cos A = cos2 – sin2 = 2 cos2 – 1 = 1 – 2 sin2 - =
- =
- Trigonometric transformations:
- Sum-to-Product Identities
- Product-to-Sum Identities
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Co-function Identities:
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Negative Angle Identities:
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Sum and Difference Identities:
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Double Angle Identities:
- =
- cot A – tan A = 2cot(2A) ⇒ cot(A/2) – tan(A/2) = 2cot A
- cot A – tan A = 2cosec(2A) ⇒ cot(A/2) – tan(A/2) = 2cosec A
- tan(θ/2).(1 + secθ) = tanθ
- tan(θ/2).(1 + secθ).(1 + sec2θ)........(1 + secn 2θ) = tann(2θ)
- (2cos2θ – 1) (2cos2θ + 1) = 2cos(2θ) + 1
- (2cosθ – 1).(2cos2θ – 1).(2cos4θ – 1) ........ ((2cosn – 12θ) – 1) =
- (cosθ).(cos(2θ)).(cos(4θ)).....cosn – 1(2θ) =
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Triple Angle Identities:
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Half Angle Identities:
- →
- →
- →
- →
- Conditional Identities:
If A + B + C = π, then- tan A + tan B + tan C = tan A . tan B . tan C
- cot B . cot C + cot C . cot A + cot A . cot B = 1
- tan . tan + tan . tan + tan . tan = 1
- cot + cot + cot = cot . cot . cot
- sin 2A + sin 2B + sin 2C = 4 sin A . sin B . sin C
- cos 2A + cos 2B + cos 2C = – 1 – 4 cos A . cos B . cos C
- cos2 A + cos2 B + cos2 C = – 1 – 4 cos A . cos B . cos C
- sin A + sin B + sin C = 4 cos . cos . cos
- cos A + cos B + cos C = 1 + 4 sin . sin . sin
- If A + B + C = π, then
- cos (A + B) = – cos C ; sin C = sin (A + B)
- tan (C + A) = – tan B ; cot A = – cot (B + C)
- cos = sin ; cos = sin
- sin = cos ; sin = cos
- tan = cot; tan = cot
Properties of trigonometrical functions
Definitions
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Trigonometric Function Fundamental Period sin x, cosec x 2π cos x, sec x 2π tan x, cot x π |sin x|, |cos x|, |tan x|, |cot x|, |sec x|, |cosec x| π |tan x + cot x| and |tan x – cot x| |sin x + cos x| and |sin x – cos x| π a|sin x| + b|cos x| and a|cosec x| + b|sec x| , if a = b, π if a ≠ b sin2n x + cos2n x, tan2n x + cot2n x; (n ∈ 2+) Trigonometric
FunctionDomain Range sin R [–1, 1] cos R [– 1, 1] tan R – {(2n + 1), n ∈ z} R cot R – {nπ, n ∈ z} R cosec R – {nπ, n ∈ z} (–∞, –1] ⋃ [1, ∞) sec R – {(2n + 1), n ∈ z} (– ∞, –1] ⋃ [1, ∞)
- The extreme values of the function of the form a cos(αx + β) + b sin(αx + β) are also and
- The extreme values of the function of the form a cos x + b sin x + c are c and c +
- The extreme values of a sin2x + b sin x cos x + c sin2x are
- a2 sec2x + b2 cosec2x ≥ (a + b)2 occurs at x = tan–1
- a2 sin2θ + b2 cosec2θ ≥ 2ab
- a2 tan2θ + b2 cot2θ ≥ 2ab
- a2 cos2θ + b2 sec2θ ≥ 2ab
- f(θ) =
Minimum value of f(θ) = a + b and occurs at θ = 0
Maximum value of f(θ) = and occurs at θ = - In a triangle ABC
(i) sin A + sin B + sin C ≤
(ii) sin (A/2) + sin (B/2) + sin (C/2) ≤
(iii) cos (A/2) + cos (B/2) + cos (C/2) ≤
(iv) tan A + tan B + tan C ≥ 3√3 and tan A tan B tan C ≥ 3√3, so cot A cot B cot C ≤ (1/3√3)
(v) tan2(A/2) + tan2(B/2) + tan2(C/2) ≥ 1
(vi) cos A + cos B + cos C ≤
(vii) - acos2x + asin2x ≥ 2√a ∀ (a > 0)
- If the range of f(x) is [a, 0], then the range of (1/f(x)) is (–∞, (1/a)], (a > 0)