Essential Trigonometric Identities and Properties

Trigonometric Functions and Values

Standard Angle Values

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

• sin(37°) = 3/5, sin(53°) = 4/5

• cos(37°) = 4/5, cos(53°) = 3/5

• tan(37°) = 3/4, tan(53°) = 4/3

Example: To solve a right triangle with one angle of 37°, use these values to find side ratios quickly.

Trigonometric Identities

Angle Addition and Subtraction Formulas

These identities allow the calculation of trigonometric functions for sums or differences of 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 $\sin(75°)$, use $\sin(45° + 30°)$ and apply the sine addition formula.

Double Angle and Power-Reducing Formulas

These formulas are used to simplify expressions involving multiple angles or powers of trigonometric functions.

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

Example: $\cos 2\theta$ can be used to express $\cos^2 \theta$ in terms of $\cos 2\theta$.

Trigonometric Function Transformations

Negative Angles and Co-function Identities

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

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

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

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

Example: $\cos(180° - \theta) = -\cos \theta$

Graphs of Trigonometric Functions

Basic Graphs and Properties

The graphs of sine, cosine, and tangent functions are periodic and have characteristic shapes, amplitudes, and periods.

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

• Tangent: Periodic with period $\pi$

• Amplitude: Maximum value for sine and cosine is 1

Example: The graph of $y = \sin \theta$ oscillates between -1 and 1.

Summary Table: Key Trigonometric Values

The following table summarizes the key values for sine, cosine, and tangent at standard angles:

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

Additional info: Table values inferred for completeness and standard reference.

Applications of Trigonometric Identities

Simplifying Expressions and Solving Equations

Trigonometric identities are essential tools for simplifying complex expressions and solving equations involving trigonometric functions.

• Use angle addition and double angle formulas to rewrite expressions in simpler forms.

• Apply standard values and identities to solve equations for unknown angles or sides in triangles.

Example: To solve $\sin(2\theta) = \sqrt{3}/2$, use the double angle formula and standard values.