Trigonometric Identities

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Basic Trigonometric Identities

  • sin2θ + cos2θ = 1 ; −1 ≤ sin θ ≤ 1 ; −1 ≤ cos θ ≤ 1 ∀ θ ∈ R
  • sec2θ − tan2θ = 1 ; |sec θ| ≥ 1 ∀ θ ∈ R
  • cosec2θ − cot2θ = 1 ; |cosec θ| ≥ 1 ∀ θ ∈ R

Important Trigonometric Ratios | Trigonometry

  • sin n π = 0 ; cos n π = (-1)n ; tan n π = 0 where n ∈ I
  • (sin frac{(2 n+1) pi}{2}=(-1)^{n} quad & quad cos frac{(2 n+1) pi}{2}=0)
  • sin 15° or (sin frac{pi}{12}=frac{sqrt{3}-1}{2 sqrt{2}}=cos 75^{circ} text { or } cos frac{5 pi}{12});
    (cos 15^{circ} text { or } cos frac{pi}{12}=frac{sqrt{3}+1}{2 sqrt{2}}=sin 75^{circ} text { or } sin frac{5 pi}{12})
    (tan 15^{circ}=frac{sqrt{3}-1}{sqrt{3}+1}=2-sqrt{3}=cot 75^{circ} ; tan 75^{circ}=frac{sqrt{3}+1}{sqrt{3}-1}=2+sqrt{3}=cot 15^{circ})
  • (sin frac{pi}{8}=frac{sqrt{2-sqrt{2}}}{2} ; quad cos frac{pi}{8}=frac{sqrt{2+sqrt{2}}}{2} ; tan frac{pi}{8}=sqrt{2}-1 ; quad tan frac{3 pi}{8}=sqrt{2}+1)
  • (sin frac{pi}{10} text { or } sin 18^{circ}=frac{sqrt{5}-1}{4} quad & quad cos 36^{circ} quad text { or } cos frac{pi}{5}=frac{sqrt{5}+1}{4})

Trigonometric Functions of Allied Angles | Trigonometric Identities

If θ is any angle, then − θ, 90 ± θ, 180 ± θ, 270 ± θ, 360 ± θ etc. are called Allied Angles.

  • sin (− θ) = − sin θ ; cos (− θ) = cos θ
  • sin (90°- θ) = cos θ ; cos (90° − θ) = sin θ
  • sin (90°+ θ) = cos θ ; cos (90°+ θ) = − sin θ
  • sin (180°− θ) = sin θ ; cos (180°− θ) = − cos θ
  • sin (180°+ θ) = − sin θ; cos (180°+ θ) = − cos θ
  • sin (270°− θ) = − cos θ ; cos (270°− θ) = − sin θ
  • sin (270°+ θ) = − cos θ ; cos (270°+ θ) = sin θ

Trigonometric Functions of Sum or Difference of Two Angles | Trigonometry

  • sin (A ± B) = sinA cosB ± cosA sinB
  • cos (A ± B) = cosA cosB ∓ sinA sinB
  • sin²A − sin²B = cos²B − cos²A = sin (A+B) . sin (A− B)
  • cos²A − sin²B = cos²B − sin²A = cos (A+B) . cos (A − B)
  • (tan (mathrm{A} pm mathrm{B})=frac{tan mathrm{A} pm tan mathrm{B}}{1 mp tan mathrm{A} tan mathrm{B}}[/latexl]
  • [latex]cot (mathrm{A} pm mathrm{B})=frac{cot mathrm{A} cot mathrm{B} mp 1}{cot mathrm{B} pm cot mathrm{A}})

Factorisation of The Sum or Difference of Two Sines or Cosines | Trigonometric Identities

  • (sin C+sin D=2 sin frac{C+D}{2} cos frac{C-D}{2})
  • (sin C-sin D=2 cos frac{C+D}{2} sin frac{C-D}{2})
  • (cos C+cos D=2 cos frac{C+D}{2} cos frac{C-D}{2})
  • (cos C-cos D=-2 sin frac{C+D}{2} sin frac{C-D}{2})

Transformation of Products Into Sum or Difference of Sines & Cosines | Trigonometric Identities

  • 2 sinA cosB = sin(A+B) + sin(A−B)
  • 2 cosA sinB = sin(A+B) − sin(A−B)
  • 2 cosA cosB = cos(A+B) + cos(A−B)
  • 2 sinA sinB = cos(A−B) − cos(A+B)

Multiple Angles And Half Angles | Trigonometric Identities

  • sin 2A = 2 sinA cosA; (sin theta=2 sin frac{theta}{2} cos frac{theta}{2})
  • cos2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2 sin²A;
    (cos theta=cos ^{2} frac{theta}{2}-sin ^{2} frac{theta}{2}=2 cos ^{2} frac{theta}{2}-1=1-2 sin ^{2} frac{theta}{2})
    2 cos²A = 1 + cos 2A , 2sin²A = 1 − cos 2A; (tan ^{2} A=frac{1-cos 2 A}{1+cos 2 A})
    (2 cos ^{2} frac{theta}{2}=1+cos theta, 2 sin ^{2} frac{theta}{2}=1-cos theta)
  • (tan 2 mathrm{A}=frac{2 tan mathrm{A}}{1-tan ^{2} mathrm{A}} quad ; quad tan theta=frac{2 tan (theta / 2)}{1-tan ^{2}(theta / 2)})
  • (sin 2 A=frac{2 tan A}{1+tan ^{2} A}, quad cos 2 A=frac{1-tan ^{2} A}{1+tan ^{2} A})
  • sin 3A = 3 sinA − 4 sin3A
  • cos 3A = 4 cos3A − 3 cosA
  • (tan 3 A=frac{3 tan A-tan ^{3} A}{1-3 tan ^{2} A})

Three Angles | Trigonometric Identities

  • (a) (tan (mathrm{A}+mathrm{B}+mathrm{C})=frac{tan mathrm{A}+tan mathrm{B}+tan mathrm{C}-tan mathrm{A} tan mathrm{B} tan mathrm{C}}{1-tan mathrm{A} tan mathrm{B}-tan mathrm{B} tan mathrm{C}-tan mathrm{C} tan mathrm{A}})
    Note If:

    • A+B+C = π then tanA + tanB + tanC = tanA tanB tanC
    • A+B+C = (frac {pi}{2}) then tanA tanB + tanB tanC + tanC tanA = 1
  • (b) If A + B + C = π then:
    • sin2A + sin2B + sin2C = 4 sinA sinB sinC
    • sinA + sinB + sinC = 4 cos(frac {A}{2}) cos(frac {B}{2}) cos(frac {C}{2})

Maximum & Minimum Values of Trigonometric Functions

  • Min. value of a2tan2θ + b2cot2θ = 2ab where θ ∈ R
  • Max. and Min. value of acosθ + bsinθ are (sqrt{a^{2}+b^{2}} text { and }-sqrt{a^{2}+b^{2}})
  • If f(θ) = acos(α + θ) + bcos(β + θ) where a, b, α and β are known quantities then –
    (sqrt{a^{2}+b^{2}+2 a b cos (alpha-beta)} leq f(theta) leq sqrt{a^{2}+b^{2}+2 a b cos (alpha-beta)})
  • If α,β ∈ (0, (frac {pi}{2})) and α + β = σ (constant) then the maximum values of the expression cosα cosβ, cosα + cosβ, sinα + sinβ and sinα sinβ occurs when α = β = σ/2.
  • If α,β ∈ (0, (frac {pi}{2})) and α + β = σ(constant) then the minimum values of the expression secα + secβ, tanα + tanβ, cosecα + cosecβ occurs when α = β = σ/2.
  • If A, B, C are the angles of a triangle then maximum value of sinA + sinB + sinC and sinA sinB sinC occurs when A = B = C = 600
  • In case a quadratic in sinθ or cosθ is given then the maximum or minimum values can be interpreted by making a perfect square.

Sum of sines or cosines of n angles,
jee maths formulas vectors 7
(begin{array}{l}{sin alpha+sin (alpha+beta)+sin (alpha+2 beta)+ldots ldots+sin (alpha+overline{n-1} beta)=frac{sin frac{n beta}{2}}{sin frac{beta}{2}} sin left(alpha+frac{n-1}{2} betaright)} \ {cos alpha+cos (alpha+beta)+cos (alpha+2 beta)+ldots ldots+cos (alpha+overline{n-1} beta)=frac{sin frac{n beta}{2}}{sin frac{beta}{frac{beta}{2}}} cos left(alpha+frac{n-1}{2} betaright)}end{array})

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