Essential Trigonometric Identities, Formulas, and Graphs

Trigonometric Functions and Values

Standard Angle Values

Trigonometric functions for specific angles are fundamental in solving problems involving triangles and periodic phenomena. The values for sine, cosine, and tangent at 37° and 53° are commonly used in trigonometry.

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

Trigonometric Identities

Sum and Difference Formulas

Sum and difference formulas allow the calculation of trigonometric functions for the sum or difference of two angles. These are essential for simplifying expressions and solving equations.

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

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

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

Double Angle and Power Reduction Formulas

Double angle formulas are used to express trigonometric functions of double angles in terms of single angles. Power reduction formulas help in expressing powers of sine and cosine in terms of multiple angles.

• Cosine Double Angle: $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$

• Sine Double Angle: $\sin 2\theta = 2\sin \theta \cos \theta$

Trigonometric Equations and Transformations

Angle Transformations

Transformations involving angles, such as $\cos(90^\circ - \theta)$ and $\cos(180^\circ - \theta)$, are useful for simplifying trigonometric expressions and solving equations.

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

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

Trigonometric Series and Slope Variation

Arithmetic Progression (AP) Series

Trigonometric series often appear in the context of arithmetic progressions, which are sequences where each term increases by a constant difference.

• General term of AP: $a_n = a + (n-1)d$

Slope Variation and Graphs

The slope of a function and its variation are important in understanding the behavior of trigonometric graphs. The slope is given by the derivative, and its sign indicates whether the function is increasing or decreasing.

• Slope positive: Function is increasing

• Slope negative: Function is decreasing

• Maxima and minima: Points where the slope changes sign

Trigonometric Graphs

Graphical Representation

Graphs of trigonometric functions such as sine and cosine illustrate their periodic nature, amplitude, and phase shift. These graphs are essential for visualizing solutions and understanding function behavior.

• Sine and Cosine Graphs: Periodic with period $2\pi$

• Amplitude: Maximum value of the function

• Phase shift: Horizontal shift of the graph

Summary Table: Trigonometric Identities

The following table summarizes key trigonometric identities and their formulas:

Identity	Formula
Sine Sum/Difference	$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
Cosine Sum/Difference	$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
Tangent Sum/Difference	$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$
Cosine Double Angle	$\cos 2\theta = \cos^2 \theta - \sin^2 \theta$
Sine Double Angle	$\sin 2\theta = 2\sin \theta \cos \theta$
Angle Transformation	$\cos(90^\circ - \theta) = \sin \theta$
Angle Transformation	$\cos(180^\circ - \theta) = -\cos \theta$

Additional info: Some formulas and values were inferred from context and standard trigonometric knowledge to ensure completeness and clarity.