Trigonometric Identities and Formulas

Vocabulary-style flashcards derived from lecture notes on trigonometric triple angle formulas, double angle identities, and value derivations.

Double Angle Formula for Cosine (in terms of cosine)

$$\cos(2A) = 2\text{cos}^2(A) - 1$$

Double Angle Formula for Cosine (in terms of sine)

$$\cos(2A) = 1 - 2\text{sin}^2(A)$$

Double Angle Formula for Sine

$$\sin(2A) = 2\text{sin}(A)\text{cos}(A)$$

Double Angle Formula for Tangent

$$\tan(2\theta) = \frac{2\tan(\theta)}{1 - \tan^2(\theta)}$$

Sine in terms of Tangent

$$\sin(2A) = \frac{2\tan(A)}{1 + \tan^2(A)}$$

Value of $\tan(15^\text{o})$

$$2 - \text{√} 3$$

Value of $\tan(22.5^\text{o})$

$$\text{√} 2 - 1$$

Sum of Fourth Powers Identity

$$\text{sin}^4(\theta) + \text{cos}^4(\theta) = 1 - 2\text{sin}^2(\theta)\text{cos}^2(\theta)$$

Sum of Sixth Powers Identity

$$\text{sin}^6(\theta) + \text{cos}^6(\theta) = 1 - 3\text{sin}^2(\theta)\text{cos}^2(\theta)$$

Componendo and Dividendo Rule

If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a+b}{a-b} = \frac{c+d}{c-d}$

Comparison of Sine and Cosine for $\theta ∈ (0, 45^\text{o})$

$$\text{cos}(\theta) > \text{sin}(\theta)$$

Triple Angle Formula for Sine

$$\sin(3\theta) = 3\text{sin}(\theta) - 4\text{sin}^3(\theta)$$

Triple Angle Formula for Cosine

$$\cos(3\theta) = 4\text{cos}^3(\theta) - 3\text{cos}(\theta)$$

Triple Angle Formula for Tangent

$$\tan(3\theta) = \frac{3\tan(\theta) - \tan^3(\theta)}{1 - 3\tan^2(\theta)}$$

Product Type Triple Angle Identity for Sine

$$\text{sin}(\theta) \times \text{sin}(60^\text{o} - \theta) \times \text{sin}(60^\text{o} + \theta) = \frac{1}{4}\text{sin}(3\theta)$$

Product Type Triple Angle Identity for Cosine

$$\text{cos}(\theta) \times \text{cos}(60^\text{o} - \theta) \times \text{cos}(60^\text{o} + \theta) = \frac{1}{4}\text{cos}(3\theta)$$

Product Type Triple Angle Identity for Tangent

$$\text{tan}(\theta) \times \text{tan}(60^\text{o} - \theta) \times \text{tan}(60^\text{o} + \theta) = \text{tan}(3\theta)$$

Sub-multiple Angle relation for Cosine

$$1 - \text{cos}(A) = 2\text{sin}^2\text{(}\frac{A}{2}\text{)}$$