Ratios & Identities: Domain & Range of Trigonometric Ratios

Trigonometric ratios are functions that relate the angles of a right-angled triangle to the ratios of its sides. These functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Understanding the domain and range of these functions is crucial for solving trigonometric equations and modeling periodic phenomena.

Domain and Range Definitions

• Domain: The set of all possible input values (usually angles) for which the function is defined.
• Range: The set of all possible output values that the function can produce.

Trigonometric Ratios and Their Domains and Ranges

Here is a table summarizing the domains and ranges of the six basic trigonometric functions:

Function	Ratio	Domain	Range
Sine (sin)	Opposite/Hypotenuse	All real numbers	[-1, 1]
Cosine (cos)	Adjacent/Hypotenuse	All real numbers	[-1, 1]
Tangent (tan)	Opposite/Adjacent	All real numbers except odd multiples of $\frac{\pi}{2}$	All real numbers
Cosecant (csc)	Hypotenuse/Opposite	All real numbers except multiples of $\pi$	(-∞, -1] ∪ [1, ∞)
Secant (sec)	Hypotenuse/Adjacent	All real numbers except odd multiples of $\frac{\pi}{2}$	(-∞, -1] ∪ [1, ∞)
Cotangent (cot)	Adjacent/Opposite	All real numbers except multiples of $\pi$	All real numbers

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the occurring variables where both sides of the identity are defined.

Fundamental Trigonometric Identities

1. Pythagorean Identities:

   • $\sin^2(x) + \cos^2(x) = 1$
   • $1 + \tan^2(x) = \sec^2(x)$
   • $1 + \cot^2(x) = \csc^2(x)$

2. Reciprocal Identities:

   • $\csc(x) = \frac{1}{\sin(x)}$
   • $\sec(x) = \frac{1}{\cos(x)}$
   • $\cot(x) = \frac{1}{\tan(x)}$

3. Quotient Identities:

   • $\tan(x) = \frac{\sin(x)}{\cos(x)}$
   • $\cot(x) = \frac{\cos(x)}{\sin(x)}$

Examples

1. Example of Domain: The function $\tan(x)$ is undefined when $\cos(x) = 0$, which occurs at odd multiples of $\frac{\pi}{2}$. Therefore, the domain of $\tan(x)$ is all real numbers except for $x = \frac{\pi}{2} + k\pi$, where $k$ is an integer.

2. Example of Range: The function $\sin(x)$ has a range of [-1, 1] because the sine of any angle cannot exceed 1 or be less than -1, as it represents the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.

3. Example of Identity Use: To prove that $\cos^2(x) = 1 - \sin^2(x)$, we can use the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$ and rearrange it to isolate $\cos^2(x)$.

Applications

Understanding the domain and range of trigonometric functions is essential for:

• Solving trigonometric equations.
• Graphing trigonometric functions.
• Modeling periodic phenomena such as waves and oscillations.
• Analyzing sound and light waves in physics.
• Engineering, especially in signal processing and control systems.

Conclusion

The domain and range of trigonometric ratios are fundamental concepts in trigonometry. They define the set of angles for which the functions are valid and the possible values these functions can output. Trigonometric identities are tools that help simplify and solve trigonometric expressions and equations. Mastery of these concepts is essential for success in mathematics, physics, engineering, and related fields.