Primary Ratios:
\( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
\( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
\( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)

Reciprocal Ratios:
\( \cosec \theta = \frac{1}{\sin \theta} = \frac{\text{Hypotenuse}}{\text{Opposite}} \)
\( \sec \theta = \frac{1}{\cos \theta} = \frac{\text{Hypotenuse}}{\text{Adjacent}} \)
\( \cot \theta = \frac{1}{\tan \theta} = \frac{\text{Adjacent}}{\text{Opposite}} \)

Quotient Relations:
\( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
\( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

For Sine:
\[ \sin 2A = 2 \sin A \cos A \]
\[ \sin 2A = \frac{2 \tan A}{1 + \tan^2 A} \]

For Cosine:
\[ \cos 2A = \cos^2 A - \sin^2 A \]
\[ \cos 2A = 2\cos^2 A - 1 \]
\[ \cos 2A = 1 - 2\sin^2 A \]
\[ \cos 2A = \frac{1 - \tan^2 A}{1 + \tan^2 A} \]

For Tangent:
\[ \tan 2A = \frac{2 \tan A}{1 - \tan^2 A} \]

Triple Angle Identities:
\[ \sin 3A = 3\sin A - 4\sin^3 A \]
\[ \cos 3A = 4\cos^3 A - 3\cos A \]
\[ \tan 3A = \frac{3\tan A - \tan^3 A}{1 - 3\tan^2 A} \]

Half Angle Identities:
\[ \sin \frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{2}} \]
\[ \cos \frac{A}{2} = \pm\sqrt{\frac{1 + \cos A}{2}} \]
\[ \tan \frac{A}{2} = \pm\sqrt{\frac{1 - \cos A}{1 + \cos A}} = \frac{\sin A}{1 + \cos A} = \frac{1 - \cos A}{\sin A} \]

Sum to Product Identities:
\[ \sin x + \sin y = 2 \sin \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right) \]
\[ \sin x - \sin y = 2 \cos \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right) \]
\[ \cos x + \cos y = 2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right) \]
\[ \cos x - \cos y = -2 \sin \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right) \]

Product to Sum Identities:
\[ \sin x \sin y = \frac{1}{2}[\cos(x-y) - \cos(x+y)] \]
\[ \cos x \cos y = \frac{1}{2}[\cos(x-y) + \cos(x+y)] \]
\[ \sin x \cos y = \frac{1}{2}[\sin(x+y) + \sin(x-y)] \]
\[ \cos x \sin y = \frac{1}{2}[\sin(x+y) - \sin(x-y)] \]

Periodic Properties:
\[ \sin(2n\pi + \theta) = \sin \theta \]
\[ \cos(2n\pi + \theta) = \cos \theta \]
\[ \tan(n\pi + \theta) = \tan \theta \]
\[ \cosec(2n\pi + \theta) = \cosec \theta \]
\[ \sec(2n\pi + \theta) = \sec \theta \]
\[ \cot(n\pi + \theta) = \cot \theta \]

Complementary Angles \((\frac{\pi}{2} - \theta)\):
\[ \sin\left(\frac{\pi}{2} - \theta\right) = \cos \theta \]
\[ \cos\left(\frac{\pi}{2} - \theta\right) = \sin \theta \]
\[ \tan\left(\frac{\pi}{2} - \theta\right) = \cot \theta \]

Supplementary Angles \((\pi - \theta)\):
\[ \sin(\pi - \theta) = \sin \theta \]
\[ \cos(\pi - \theta) = -\cos \theta \]
\[ \tan(\pi - \theta) = -\tan \theta \]

Angles \((\pi + \theta)\):
\[ \sin(\pi + \theta) = -\sin \theta \]
\[ \cos(\pi + \theta) = -\cos \theta \]
\[ \tan(\pi + \theta) = \tan \theta \]

Angles \((2\pi - \theta)\):
\[ \sin(2\pi - \theta) = -\sin \theta \]
\[ \cos(2\pi - \theta) = \cos \theta \]
\[ \tan(2\pi - \theta) = -\tan \theta \]

a sin θ + b cos θ Form:
\[ a \sin \theta + b \cos \theta = \sqrt{a^2 + b^2} \sin(\theta + \alpha) \]
\[ \text{where } \tan \alpha = \frac{b}{a} \]

\[ a \sin \theta + b \cos \theta = \sqrt{a^2 + b^2} \cos(\theta - \beta) \]
\[ \text{where } \tan \beta = \frac{a}{b} \]