Essential Trigonometric Identities and Properties

Trigonometric Functions and Standard Values

Key Trigonometric Ratios

Trigonometric functions relate the angles of a triangle to the ratios of its sides. The primary trigonometric functions are sine, cosine, and tangent, commonly abbreviated as sin, cos, and tan, respectively. For certain standard angles, these functions have well-known values that are frequently used in calculations.

• sin(37°) = \frac{3}{5}

• sin(53°) = \frac{4}{5}

• cos(37°) = \frac{4}{5}

• cos(53°) = \frac{3}{5}

• tan(37°) = \frac{3}{4}

• tan(53°) = \frac{4}{3}

Example: To find the sine of 37°, use the value above: $\sin(37^\circ) = \frac{3}{5}$.

Trigonometric Identities

Angle Addition and Subtraction Formulas

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. The angle addition and subtraction formulas are fundamental for simplifying expressions and solving equations involving trigonometric functions.

• 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 the addition formula.

Double Angle and Power-Reducing Formulas

Double Angle Formulas

Double angle formulas are used to express trigonometric functions of double angles (2θ) in terms of single angles (θ). These are useful for simplifying expressions and solving equations.

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

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

Example: $\cos 2\theta$ can be written as $1 - 2\sin^2 \theta$ or $2\cos^2 \theta - 1$ depending on the context.

Trigonometric Function Transformations

Negative Angles and Co-function Identities

Trigonometric functions have specific properties when their arguments are negative or shifted by certain angles. These properties are called co-function and even-odd identities.

• Sine: $\sin(-\theta) = -\sin \theta$

• Cosine: $\cos(-\theta) = \cos \theta$

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

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

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 a period of $2\pi$ and range from -1 to 1.

• Tangent: Has a period of $\pi$ and is undefined at odd multiples of $\frac{\pi}{2}$.

Example: The graph of $y = \sin x$ starts at 0, reaches 1 at $\frac{\pi}{2}$, 0 at $\pi$, -1 at $\frac{3\pi}{2}$, and returns to 0 at $2\pi$.

Summary Table: Standard Trigonometric Values

The following table summarizes the standard values for sine, cosine, and tangent for commonly used angles:

Angle (°)	sin	cos	tan
0	0	1	0
30	1/2	\sqrt{3}/2	1/\sqrt{3}
37	3/5	4/5	3/4
45	1/\sqrt{2}	1/\sqrt{2}	1
53	4/5	3/5	4/3
60	\sqrt{3}/2	1/2	\sqrt{3}
90	1	0	Undefined

Additional info: Some images and formulas in the source also reference logarithms and calculus concepts, but only trigonometric content is included here as per the course relevance.