Solve over the Interval -tan(x)-sec(x)=1 , [0,2pi)
Sure, the problem you provided is a trigonometric equation where you are being asked to find all the solutions for the variable x, within the interval from 0 to 2π (not including 2π), that satisfy the equation -tan(x) - sec(x) = 1. Here, tan(x) and sec(x) are trigonometric functions representing the tangent and the secant of angle x, respectively. The equation needs to be solved for x such that each solution falls within the given interval, which is one full cycle around the unit circle for trigonometric functions.
$- tan \left(\right. x \left.\right) - sec \left(\right. x \left.\right) = 1$,$\left[\right. 0 , 2 \pi \left.\right)$
Answer
Solution:
Step:1
To find the solution, graph both sides of the equation separately. The x-values where the graphs intersect represent the solutions. The general solution is $x = \pi + 2 \pi n$, where $n$ is any integer.
Step:2
Determine if the general solution falls within the interval $[0, 2\pi)$ by substituting $n=0$.
Step:2.1
Substitute $n=0$ into the general solution: $\pi + 2 \pi (0)$.
Step:2.2
Proceed to simplify the expression.
Step:2.2.1
Calculate $2 \pi (0)$.
Step:2.2.1.1
Perform the multiplication of $0$ and $2$: $\pi + 0 \pi$.
Step:2.2.1.2
Multiply $0$ by $\pi$: $\pi + 0$ which simplifies to $\pi + 0$.
Step:2.2.2
Combine $\pi$ and $0$: $\pi$.
Step:2.3
Verify that $\pi$ is within the given interval $[0, 2\pi)$. The solution is $x = \pi$.
Step:3
There is no further action required in this step as the solution has been found.
Knowledge Notes:
Trigonometric Functions: The tangent and secant are trigonometric functions. The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side, and the secant is the reciprocal of the cosine function.
Graphical Solution: Graphing functions can provide a visual representation of solutions, especially when solving trigonometric equations. The points of intersection between the graphs of two functions correspond to the x-values where the functions are equal.
Interval Notation: The interval $[0, 2\pi)$ includes all numbers from $0$ to $2\pi$, including $0$ but excluding $2\pi$. This is common in trigonometric problems where the domain is often restricted to a specific range of angles.
General Solution for Trigonometric Equations: Trigonometric equations often have an infinite number of solutions that can be expressed in a general form, such as $x = \pi + 2 \pi n$, where $n$ is an integer. This accounts for the periodic nature of trigonometric functions.
LaTeX Formatting: In the solution, LaTeX is used to format mathematical expressions. For example, $\pi + 2 \pi (0)$ is rendered in LaTeX to display the mathematical symbols and operations clearly.
Simplification: The process of simplifying expressions involves performing arithmetic operations and combining like terms to rewrite the expression in a more basic or concise form.
