📌 Basic Trigonometry

Sine:

\( \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \)

Cosine:

\( \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)

Tangent:

\( \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \)

Cosecant:

\( \csc \theta = \frac{1}{\sin \theta} \)

Secant:

\( \sec \theta = \frac{1}{\cos \theta} \)

Cotangent:

\( \cot \theta = \frac{1}{\tan \theta} \)

Pythagorean Identity:

\( \sin^2 \theta + \cos^2 \theta = 1 \)

Angle Sum Identity (Sine):

\( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

Angle Sum Identity (Cosine):

\( \cos(A + B) = \cos A \cos B - \sin A \sin B \)

Angle Sum Identity (Tangent):

\( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)

Double Angle Identity (Sine):

\( \sin 2\theta = 2 \sin \theta \cos \theta \)

Double Angle Identity (Cosine):

\( \cos 2\theta = \cos^2 \theta - \sin^2 \theta \)

Double Angle Identity (Tangent):

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

Half Angle Identity (Sine):

\( \sin \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{2}} \)

Half Angle Identity (Cosine):

\( \cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos \theta}{2}} \)

Half Angle Identity (Tangent):

\( \tan \frac{\theta}{2} = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} \)

📌 Trigonometric Identities

Reciprocal Identities:

\( \csc \theta = \frac{1}{\sin \theta} \), \( \sec \theta = \frac{1}{\cos \theta} \), \( \cot \theta = \frac{1}{\tan \theta} \)

Quotient Identities:

\( \tan \theta = \frac{\sin \theta}{\cos \theta} \), \( \cot \theta = \frac{\cos \theta}{\sin \theta} \)

Pythagorean Identities:

\( \sin^2 \theta + \cos^2 \theta = 1 \), \( 1 + \tan^2 \theta = \sec^2 \theta \), \( 1 + \cot^2 \theta = \csc^2 \theta \)

Even-Odd Identities:

\( \sin(-\theta) = -\sin \theta \), \( \cos(-\theta) = \cos \theta \), \( \tan(-\theta) = -\tan \theta \)

Sum-to-Product Identities:

\( \sin A + \sin B = 2 \sin \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \), \( \sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \), \( \cos A + \cos B = 2 \cos \left( \frac{A + B}{2} \right) \cos \left( \frac{A - B}{2} \right) \), \( \cos A - \cos B = -2 \sin \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right) \)

Product-to-Sum Identities:

\( \sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \), \( \cos A \cos B = \frac{1}{2} [\cos(A - B) + \cos(A + B)] \), \( \sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \)

📌 Inverse Trigonometry

Inverse Sine:

\( \sin^{-1} x = \theta \), where \( \sin \theta = x \)

Inverse Cosine:

\( \cos^{-1} x = \theta \), where \( \cos \theta = x \)

Inverse Tangent:

\( \tan^{-1} x = \theta \), where \( \tan \theta = x \)

Inverse Cosecant:

\( \csc^{-1} x = \sin^{-1} \left( \frac{1}{x} \right) \)

Inverse Secant:

\( \sec^{-1} x = \cos^{-1} \left( \frac{1}{x} \right) \)

Inverse Cotangent:

\( \cot^{-1} x = \tan^{-1} \left( \frac{1}{x} \right) \)

Range of Inverse Sine:

\( [-\frac{\pi}{2}, \frac{\pi}{2}] \)

Range of Inverse Cosine:

\( [0, \pi] \)

Range of Inverse Tangent:

\( (-\frac{\pi}{2}, \frac{\pi}{2}) \)

Derivative of Inverse Sine:

\( \frac{d}{dx} \sin^{-1} x = \frac{1}{\sqrt{1 - x^2}} \)

Derivative of Inverse Cosine:

\( \frac{d}{dx} \cos^{-1} x = -\frac{1}{\sqrt{1 - x^2}} \)

Derivative of Inverse Tangent:

\( \frac{d}{dx} \tan^{-1} x = \frac{1}{1 + x^2} \)

📌 Trigonometric Equations

General Solution for Sine:

\( \sin \theta = k \Rightarrow \theta = n\pi + (-1)^n \alpha \)

General Solution for Cosine:

\( \cos \theta = k \Rightarrow \theta = 2n\pi \pm \alpha \)

General Solution for Tangent:

\( \tan \theta = k \Rightarrow \theta = n\pi + \alpha \)

Quadratic Trigonometric Equations:

Solve \( a \sin^2 \theta + b \sin \theta + c = 0 \)

Linear Trigonometric Equations:

Solve \( a \sin \theta + b \cos \theta = c \)

📌 Applications

Law of Sines:

\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

Law of Cosines:

\( c^2 = a^2 + b^2 - 2ab \cos C \)

Area of Triangle:

\( A = \frac{1}{2} ab \sin C \)

Polar Coordinates:

\( x = r \cos \theta \), \( y = r \sin \theta \)

Complex Numbers:

\( z = r (\cos \theta + i \sin \theta) \)

Fourier Series:

\( f(x) = a_0 + \sum_{n=1}^\infty (a_n \cos nx + b_n \sin nx) \)