Trigonometric Ratios and Identities

Trigonometric Ratios

Definitions

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  The trigonometric ratios of angle θ are
  

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Formulae

• Reciprocal Trigonometric Ratios:
  

  • csc θ = or
  • sec θ = or
  • cot θ = or
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• Trigonometric-ratios of allied (or related) angles
  

  
• T-ratios of other allied angles in terms of theta:
  

  

Trigonometric Identities

Formulae

• Reciprocal trigonometric identities:
  
• 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
• 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
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  • 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
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• 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
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  • 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
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  • 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:
  

  • =
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  • 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) ........ ((2cos^{n – 1}2θ) – 1) =
  • (cosθ).(cos(2θ)).(cos(4θ)).....cos^{n – 1}(2θ) =
• Triple Angle Identities:
  

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• Half Angle Identities:
  

  • →
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  • →
  • →
• 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

• 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 Function	Domain	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, ∞)

Maximum and minimum values (or) Extreme Values

• 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)
• a^cos2x + a^sin2x ≥ 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)