# Generalized Trigonometric Intervals

• Cosmophile
In summary, the function ##f(x) = \sin x + \cos x## is increasing on the intervals ##(0, \frac {\pi}{4})## and ##(\frac {5 \pi}{4}, 2\pi)##, and is decreasing on the intervals ##(\frac {\pi}{4}, \frac {5 \pi}{4})##, with a generalization of ##f## being decreasing on all intervals of the form ##(\frac{\pi}{4} + 2\pi n, \frac{5\pi}{4} + 2\pi n)## or ##\bigcup_{n\in \mathbb{Z}} (\frac{\pi}{4} +
Cosmophile

## Homework Statement

Identify the intervals of increase/decrease of ##f(x) = \sin x + \cos x##

## Homework Equations

##f(x) = \sin x + \cos x##

##f'(x) = \cos x - \sin x = \sqrt 2 \cos(x+\frac {\pi}{4})##

## The Attempt at a Solution

##f## is increasing when ##f'(x) > 0##

##\sqrt 2 \cos(x+\frac {\pi}{4}) > 0 \to \cos(x + \frac {\pi}{4}) > 0##

So, ##f## is increasing on the intervals ##(0, \frac {\pi}{4})## and ##(\frac {5 \pi}{4}, 2\pi)##, and is decreasing on the interval ##(\frac {\pi}{4}, \frac {5 \pi}{4})##.

I know that in order to generalize this, you would add ##2 \pi n## to all intervals. I simply would like to know how this would be mathematically written. Thanks!

Like this: ##f## is decreasing on all intervals of the form ##(\frac{\pi}{4} + 2\pi n, \frac{5\pi}{4} + 2\pi n)## for each ##n\in \mathbb{Z}##.

A more compact notation is available from set theory as

$$\bigcup_{n\in \mathbb{Z}} (\frac{\pi}{4} + 2\pi n, \frac{5\pi}{4} + 2\pi n)$$

## 1. What is a generalized trigonometric interval?

A generalized trigonometric interval is a mathematical concept that involves the use of trigonometric functions, such as sine, cosine, and tangent, to represent the range of values between two points on a circle or line. It is a way to measure angles and distances in a standardized way.

## 2. How is a generalized trigonometric interval different from a regular trigonometric interval?

A regular trigonometric interval only considers the values of the trigonometric functions within a specific range, such as 0 to 360 degrees. A generalized trigonometric interval allows for a wider range of values, including negative angles and larger magnitudes, making it more versatile and applicable to a variety of mathematical problems.

## 3. What are some real-world applications of generalized trigonometric intervals?

Generalized trigonometric intervals are commonly used in fields such as engineering, physics, and astronomy to calculate distances, angles, and trajectories. They are also used in navigation systems, robotics, and computer graphics.

## 4. How do you calculate a generalized trigonometric interval?

To calculate a generalized trigonometric interval, you first need to determine the starting and ending points, as well as the direction of measurement (clockwise or counterclockwise). Then, you can use the appropriate trigonometric function to find the values for each point and calculate the difference between them.

## 5. Can generalized trigonometric intervals be used with non-right triangles?

Yes, generalized trigonometric intervals can be used with any type of triangle, including non-right triangles. In these cases, the trigonometric functions used may differ, but the concept of measuring intervals between two points remains the same.

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