Essential Trigonometric Identities and Properties

Trigonometric Functions and Their Values

Standard Angle Values

Trigonometric functions such as sine, cosine, and tangent have well-known values for specific angles. These values are fundamental for solving trigonometric equations and simplifying expressions.

• Sine Values: $\sin 37^\circ = \frac{3}{5}$, $\sin 53^\circ = \frac{4}{5}$

• Cosine Values: $\cos 37^\circ = \frac{4}{5}$, $\cos 53^\circ = \frac{3}{5}$

• Tangent Values: $\tan 37^\circ = \frac{3}{4}$, $\tan 53^\circ = \frac{4}{3}$

Example: These values are often used in right triangle problems and in simplifying trigonometric expressions.

Trigonometric Identities

Angle Addition and Subtraction Formulas

These identities allow the calculation of trigonometric functions for the sum or difference of two angles.

• Sine Addition: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

• Cosine Addition: $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

• Tangent Addition: $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Example: To find $\sin(75^\circ)$, use $\sin(45^\circ + 30^\circ)$ and apply the addition formula.

Double Angle and Power-Reducing Formulas

Double Angle Formulas

These formulas express trigonometric functions of double angles in terms of single angles.

• $\sin 2\theta = 2 \sin \theta \cos \theta$

• $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$

Example: $\cos 2\theta$ can be used to simplify expressions involving $\cos^2 \theta$ or $\sin^2 \theta$.

Trigonometric Function Transformations

Negative Angles and Co-function Identities

Trigonometric functions have specific properties when their arguments are negative or shifted by certain angles.

• $\sin(-\theta) = -\sin \theta$

• $\cos(-\theta) = \cos \theta$

• $\sin(90^\circ - \theta) = \cos \theta$

• $\cos(90^\circ - \theta) = \sin \theta$

• $\sin(180^\circ - \theta) = \sin \theta$

• $\cos(180^\circ - \theta) = -\cos \theta$

Example: $\sin(90^\circ - 30^\circ) = \cos 30^\circ$

Graphs of Trigonometric Functions

Basic Graphs and Properties

The graphs of sine, cosine, and tangent functions are periodic and have characteristic shapes. Understanding these graphs is essential for analyzing trigonometric equations and modeling periodic phenomena.

• Sine and Cosine: Both have amplitude 1 and period $2\pi$.

• Tangent: Has period $\pi$ and vertical asymptotes where $\cos \theta = 0$.

Example: The graph of $y = \sin \theta$ oscillates between -1 and 1, repeating every $2\pi$.

Summary Table: Key Trigonometric Identities

Identity	Formula	Example
Sine Addition	$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$	$\sin(45^\circ + 30^\circ)$
Cosine Addition	$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$	$\cos(60^\circ - 30^\circ)$
Tangent Addition	$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$	$\tan(45^\circ + 30^\circ)$
Double Angle (Sine)	$\sin 2\theta = 2 \sin \theta \cos \theta$	$\sin 2 \times 30^\circ$
Double Angle (Cosine)	$\cos 2\theta = 2\cos^2 \theta - 1$	$\cos 2 \times 45^\circ$

Additional info: Some content in the images relates to logarithms and calculus (e.g., slope, maxima/minima), which are not directly part of the core trigonometry curriculum. Only the trigonometric content is included here.