Trigonometry-1: Essential Identities and Problem Solving

Quadrant Signs of Trigonometric Functions

  • The location of an angle θ\theta is determined by the signs of its trigonometric ratios:
    • If sin⁡(θ)>0\sin(\theta) > 0 and sec⁡(θ)<0\sec(\theta) < 0, then θ\theta lies in the II Quadrant.
    • If sec⁡(θ)>0\sec(\theta) > 0 and cosec(θ)<0\text{cosec}(\theta) < 0, then θ\theta lies in the IV Quadrancy.
    • For π/2<θ<π\pi/2 < \theta < \pi (II Quadrant), with sin⁡(θ)=35\sin(\theta) = \frac{3}{5}, the ascending order of values is cos⁡(θ),tan⁡(θ),sin⁡(θ)\cos(\theta), \tan(\theta), \sin(\theta).

Core Formulas and Evaluation

  • Sum of squared functions at standard angles:
    • sin⁡2(0∘)+sin⁡2(π6)+sin⁡2(π3)+sin⁡2(π2)=2\sin^2(0^\circ) + \sin^2\left(\frac{\pi}{6}\right) + \sin^2\left(\frac{\pi}{3}\right) + \sin^2\left(\frac{\pi}{2}\right) = 2
    • cot⁡2(60∘)+sin⁡2(45∘)+sin⁡2(30∘)+cos⁡2(90∘)=1312\cot^2(60^\circ) + \sin^2(45^\circ) + \sin^2(30^\circ) + \cos^2(90^\circ) = \frac{13}{12}
  • The value of the expression 3cosec(20∘)−sec⁡(20∘)\sqrt{3} \text{cosec}(20^\circ) - \sec(20^\circ) is exactly 44.
  • Relationship between sec⁡(A)\sec(A) and tan⁡(A)\tan(A): If sec⁡(A)+tan⁡(A)=3\sec(A) + \tan(A) = 3, then sec⁡(A)=53\sec(A) = \frac{5}{3}.

Intervals and Extremum Values

  • Range of A=sin⁡2(θ)+cos⁡4(θ)\mathbf{A = \sin^2(\theta) + \cos^4(\theta)}: For all values of θ\theta, the range is 34≤A≤1\frac{3}{4} \le A \le 1.
  • Interval of linear combinations: The expression 3sin⁡(x)+4cos⁡(x)−13 \sin(x) + 4 \cos(x) - 1 lies within the interval [−6,4][-6, 4].
  • Comparison sign: If 0<x<π40 < x < \frac{\pi}{4}, the value of (sin⁡(x)−cos⁡(x))(\sin(x) - \cos(x)) is negative.

Simplification Identities

  • Sum-to-Product Relationships:
    • tan⁡(x)+cot⁡(x)=sec⁡(x)cosec(x)\tan(x) + \cot(x) = \sec(x) \text{cosec}(x)
    • sec⁡2(x)+cosec2(x)=sec⁡2(x)cosec2(x)\sec^2(x) + \text{cosec}^2(x) = \sec^2(x) \text{cosec}^2(x)
    • cos⁡2(x)+cos⁡4(x)=1\cos^2(x) + \cos^4(x) = 1 if sin⁡(x)+sin⁡2(x)=1\sin(x) + \sin^2(x) = 1
  • Higher Power Identities:
    • sec⁡4(x)−sec⁡2(x)=tan⁡4(x)+tan⁡2(x)\sec^4(x) - \sec^2(x) = \tan^4(x) + \tan^2(x)
    • cosec4(x)−cosec2(x)=cot⁡4(x)+cot⁡2(x)\text{cosec}^4(x) - \text{cosec}^2(x) = \cot^4(x) + \cot^2(x)
    • sec⁡6(x)−tan⁡6(x)−3sec⁡2(x)tan⁡2(x)=1\sec^6(x) - \tan^6(x) - 3 \sec^2(x) \tan^2(x) = 1

Specialized Conditional Identities

  • If tan⁡(θ)+sin⁡(θ)=m\tan(\theta) + \sin(\theta) = m and tan⁡(θ)−sin⁡(θ)=n\tan(\theta) - \sin(\theta) = n, then the identity is m2−n2=±4mnm^2 - n^2 = \pm 4 \sqrt{mn}.
  • If cos⁡(θ)+sin⁡(θ)=2cos⁡(θ)\cos(\theta) + \sin(\theta) = \sqrt{2} \cos(\theta), then the difference cos⁡(θ)−sin⁡(θ)=2sin⁡(θ)\cos(\theta) - \sin(\theta) = \sqrt{2} \sin(\theta).
  • In the identity (sin⁡(θ)+cosec(θ))2+(cos⁡(θ)+sec⁡(θ))2=K+tan⁡2(θ)+cot⁡2(θ)(\sin(\theta) + \text{cosec}(\theta))^2 + (\cos(\theta) + \sec(\theta))^2 = K + \tan^2(\theta) + \cot^2(\theta), the constant K=7K = 7.

The location of an angle θ\theta is determined by the signs of its trigonometric ratios:

  • If \sin(\theta) > 0 and \sec(\theta) > 0: Quadrant I
  • If \sin(\theta) > 0 and \sec(\theta) < 0: Quadrant II
  • If \sin(\theta) < 0 and \sec(\theta) < 0: Quadrant III
  • If sin⁡(θ)<0\sin(\theta) < 0 and sec⁡(θ)>0\sec(\theta) > 0: Quadrant IV
Core Formulas and Evaluation
  • Sum of squared functions at standard angles:
    • sin⁡2(0∘)+sin⁡2(π6)+sin⁡2(π3)+sin⁡2(π2)=2\sin^2(0^\circ) + \sin^2\left(\frac{\pi}{6}\right) + \sin^2\left(\frac{\pi}{3}\right) + \sin^2\left(\frac{\pi}{2}\right) = 2
    • cot⁡2(60∘)+sin⁡2(45∘)+sin⁡2(30∘)+cos⁡2(90∘)=1312\cot^2(60^\circ) + \sin^2(45^\circ) + \sin^2(30^\circ) + \cos^2(90^\circ) = \frac{13}{12}
  • The value of the expression 3cosec(20∘)−sec⁡(20∘)\sqrt{3} \text{cosec}(20^\circ) - \sec(20^\circ) is exactly 44.
  • Relationship between sec⁡(A)\sec(A) and tan⁡(A)\tan(A): If sec⁡(A)+tan⁡(A)=3\sec(A) + \tan(A) = 3, then sec⁡(A)=53\sec(A) = \frac{5}{3}.
Intervals and Extremum Values
  • Range of A=sin⁡2(θ)+cos⁡4(θ)\mathbf{A = \sin^2(\theta) + \cos^4(\theta)}: For all values of θ\theta, the range is 34≤A≤1\frac{3}{4} \le A \le 1.
  • Interval of linear combinations: The expression 3sin⁡(x)+4cos⁡(x)−13 \sin(x) + 4 \cos(x) - 1 lies within the interval [−6,4][-6, 4].
Simplification Identities
  • Sum-to-Product Relationships:
    • tan⁡(x)+cot⁡(x)=sec⁡(x)cosec(x)\tan(x) + \cot(x) = \sec(x) \text{cosec}(x)
    • sec⁡2(x)+cosec2(x)=sec⁡2(x)cosec2(x)\sec^2(x) + \text{cosec}^2(x) = \sec^2(x) \text{cosec}^2(x)
    • cos⁡2(x)+cos⁡4(x)=1\cos^2(x) + \cos^4(x) = 1 if sin⁡(x)+sin⁡2(x)=1\sin(x) + \sin^2(x) = 1.
  • Higher Power Identities:
    • sec⁡4(x)−sec⁡2(x)=tan⁡4(x)+tan⁡2(x)\sec^4(x) - \sec^2(x) = \tan^4(x) + \tan^2(x)
    • cosec4(x)−cosec2(x)=cot⁡4(x)+cot⁡2(x)\text{cosec}^4(x) - \text{cosec}^2(x) = \cot^4(x) + \cot^2(x)
    • sec⁡6(x)−tan⁡6(x)−3sec⁡2(x)tan⁡2(x)=1\sec^6(x) - \tan^6(x) - 3 \sec^2(x) \tan^2(x) = 1.
Specialized Conditional Identities
  • If tan⁡(θ)+sin⁡(θ)=m\tan(\theta) + \sin(\theta) = m and tan⁡(θ)−sin⁡(θ)=n\tan(\theta) - \sin(\theta) = n, then the identity is m2−n2=±4mnm^2 - n^2 = \pm 4 \sqrt{mn}.
  • If cos⁡(θ)+sin⁡(θ)=2cos⁡(θ)\cos(\theta) + \sin(\theta) = \sqrt{2} \cos(\theta), then the difference cos⁡(θ)−sin⁡(θ)=2sin⁡(θ)\cos(\theta) - \sin(\theta) = \sqrt{2} \sin(\theta).
  • In the identity (sin⁡(θ)+cosec(θ))2+(cos⁡(θ)+sec⁡(θ))2=K+tan⁡2(θ)+cot⁡2(θ)(\sin(\theta) + \text{cosec}(\theta))^2 + (\cos(\theta) + \sec(\theta))^2 = K + \tan^2(\theta) + \cot^2(\theta), the constant K=7K = 7.