Finding the range in trigonometry involves identifying all possible output values (y-values) that a trigonometric function can produce. This understanding is crucial as the range dictates the vertical span of the function's graph.
Understanding the Range of Trigonometric Functions
The range of a trigonometric function is the set of all real numbers that are results of the function for its given input values (domain). Unlike algebraic polynomials that can often span all real numbers, trigonometric functions often have restricted ranges due to their periodic nature and definitions.
To effectively determine the range, it's essential to understand each function's unique behavior, including its relationship to other functions through identities. For instance, reciprocal identities like csc x = 1/sin x and sec x = 1/cos x, along with quotient identities such as tan x = sin x / cos x and cot x = cos x / sin x, are foundational in understanding why certain output values are possible or excluded.
Range of Primary Trigonometric Functions
Let's explore the range for each of the six basic trigonometric functions:
Sine Function (sin x)
The sine function oscillates between -1 and 1. This means its maximum output value is 1 and its minimum is -1.
- Range: $[-1, 1]$
Cosine Function (cos x)
Similar to the sine function, the cosine function also oscillates between -1 and 1.
- Range: $[-1, 1]$
Tangent Function (tan x)
Defined as sin x / cos x, the tangent function can take any real value. Its graph has vertical asymptotes wherever cos x = 0, meaning it extends infinitely in both positive and negative directions between these asymptotes.
- Range: $(-\infty, \infty)$ or $\mathbb{R}$
Range of Reciprocal Trigonometric Functions
The ranges of the reciprocal functions are derived directly from the ranges of their primary counterparts.
Cosecant Function (csc x)
As the reciprocal of sine, csc x = 1/sin x. Since sin x is always between -1 and 1 (inclusive), its reciprocal, 1/sin x, can never be a value strictly between -1 and 1 (excluding 0, where csc x is undefined).
- Range: $(-\infty, -1] \cup [1, \infty)$
Secant Function (sec x)
Being the reciprocal of cosine, sec x = 1/cos x. Similar to cosecant, because cos x is between -1 and 1, 1/cos x will never fall strictly between -1 and 1 (excluding 0, where sec x is undefined).
- Range: $(-\infty, -1] \cup [1, \infty)$
Cotangent Function (cot x)
As the reciprocal of tangent, cot x = cos x / sin x. Like the tangent function, the cotangent function also spans all real numbers, with vertical asymptotes wherever sin x = 0.
- Range: $(-\infty, \infty)$ or $\mathbb{R}$
Summary of Trigonometric Function Ranges
Here's a quick reference table for the domain and range of the basic trigonometric functions:
| Function | Domain | Range |
|---|---|---|
| sin x | $(-\infty, \infty)$ | $[-1, 1]$ |
| cos x | $(-\infty, \infty)$ | $[-1, 1]$ |
| tan x | $x \neq \frac{\pi}{2} + n\pi$, $n \in \mathbb{Z}$ | $(-\infty, \infty)$ |
| csc x | $x \neq n\pi$, $n \in \mathbb{Z}$ | $(-\infty, -1] \cup [1, \infty)$ |
| sec x | $x \neq \frac{\pi}{2} + n\pi$, $n \in \mathbb{Z}$ | $(-\infty, -1] \cup [1, \infty)$ |
| cot x | $x \neq n\pi$, $n \in \mathbb{Z}$ | $(-\infty, \infty)$ |
Impact of Transformations on Range
The range of trigonometric functions can be altered by various transformations:
- Amplitude (Vertical Stretch/Compression): For functions like $y = A \sin(Bx + C) + D$ or $y = A \cos(Bx + C) + D$, the value of A (amplitude) stretches or compresses the graph vertically. The range for sine and cosine becomes $[-|A|, |A|]$.
- Vertical Shift: The value of D shifts the entire graph up or down. For sine and cosine, the range becomes $[D - |A|, D + |A|]$.
- No effect on tangent/cotangent: Amplitude and vertical shifts do not change the $(-\infty, \infty)$ range of tangent and cotangent functions, as they already span all real numbers.
- Effect on secant/cosecant: For secant and cosecant, vertical shifts and amplitude changes will affect the boundaries of their range, shifting where the values start/end relative to the original -1 and 1.
Practical Examples of Finding Range with Transformations
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Example: Find the range of $y = 3 \sin(x) + 2$
- The base sine function has a range of $[-1, 1]$.
- The amplitude is $A = 3$, so the function is stretched vertically by a factor of 3. This changes the range to $[3 \times (-1), 3 \times 1] = [-3, 3]$.
- The vertical shift is $D = 2$. We add 2 to each end of the scaled range.
- New Range: $[-3 + 2, 3 + 2] = [-1, 5]$.
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Example: Find the range of $y = -2 \cos(4x) - 1$
- The base cosine function has a range of $[-1, 1]$.
- The amplitude is $|A| = |-2| = 2$. The range becomes $[2 \times (-1), 2 \times 1] = [-2, 2]$. (The negative sign only flips the graph vertically, not the span of its values).
- The vertical shift is $D = -1$. We subtract 1 from each end of the scaled range.
- New Range: $[-2 - 1, 2 - 1] = [-3, 1]$.
By understanding the inherent limitations of each trigonometric function and how transformations affect these boundaries, you can accurately determine the range for a wide variety of trigonometric expressions.
