Question

Let $\text{Missing left or extra right}$ The sum of all distinct solutions of the equation $\sqrt{3}\sec x+\mathrm{cosec}x+2(\tan x-\cot x)=0$ in the set $S$ is equal to (A) $-\frac{7\pi }{9}$ (B) $-\frac{2\pi }{9}$ (C) 0 (D) $\frac{5\pi }{9}$

Short answer

0

Step-by-step solution

Rewrite the equation in terms of sine and cosine

Replace the trigonometric functions with their sine and cosine equivalents: $\sqrt{3}\sec x=\sqrt{3}\frac{1}{\cos x}$ and $\mathrm{cosec}x=\frac{1}{\sin x}$, $\tan x=\frac{\sin x}{\cos x}$ and $\cot x=\frac{\cos x}{\sin x}$.

Combine the terms

Combine the terms over a common denominator to simplify the equation: $\sqrt{3}\frac{1}{\cos x}+\frac{1}{\sin x}+2(\frac{\sin x}{\cos x}-\frac{\cos x}{\sin x})=0$, which simplifies to $\frac{\sqrt{3}\sin x+\cos x+2({\sin }^{2}x-{\cos }^{2}x)}{\sin x\cos x}=0$.

Solve the numerator being zero

Set the numerator equal to zero: $\sqrt{3}\sin x+\cos x+2({\sin }^{2}x-{\cos }^{2}x)=0$. This is now a quadratic trigonometric equation to be solved.

Apply the Pythagorean identity

Use the identity ${\sin }^{2}x+{\cos }^{2}x=1$ to simplify the equation further: $\sqrt{3}\sin x+\cos x+2(1-2{\cos }^{2}x)=0$.

Rearrange and solve the quadratic equation

The simplified equation $2{\cos }^{2}x-\sqrt{3}\sin x-\cos x-2=0$ is a quadratic in terms of $\cos x$ and $\sin x$. We need to solve for the values of $x$ that satisfy this equation.

Find solutions within the given set S

Find the solutions for $x$ that are in the interval $(-\pi ,\pi )$ and are distinct, not equal to $0,\pm \frac{\pi }{2}$.

Sum the solutions

Since the task requires to find the sum of all distinct solutions, once the solutions are found, they should be added together to find the total sum.

Key concepts

Secant and Cosecant Functions

Understanding secant (\text{sec}) and cosecant (\text{cosec}) functions is crucial when solving trigonometric equations. Both functions are reciprocal to basic trigonometric functions. The secant function is the reciprocal of the cosine function, where the secant of an angle in a right-angled triangle is the ratio of the hypotenuse to the adjacent side: $$\sec x=\frac{1}{\cos x}\text{.}$$ Similarly, the cosecant function is the reciprocal of the sine function, reflecting the ratio of the length of the hypotenuse to the opposite side: $$\text{cosec}x=\frac{1}{\sin x}\text{.}$$ These functions can be graphed as curves on the Cartesian plane and are periodic, like sine and cosine, but their graphs have different shapes and asymptotes.
In solving trigonometric equations, recasting secant and cosecant to their sine and cosine counterparts often simplifies the equation, facilitating further steps in the solution process.

In the exercise given, rewriting the secant and cosecant functions in terms of sine and cosine is the first step to making the equation more approachable. By understanding these relationships, you can manipulate the equation into a form where more familiar techniques, such as finding common denominators, can be applied.

Pythagorean Trigonometric Identity

A cornerstone of trigonometry is the Pythagorean trigonometric identity which is derived from the Pythagorean theorem. It expresses a fundamental relation between the sine and cosine of an angle. The identity is: $${\sin }^{2}x+{\cos }^{2}x=1\text{.}$$ This identity holds for all values of $x$.
Utilizing this identity is a powerful tool in the simplification of trigonometric equations. By substituting ${\sin }^{2}x$ or ${\cos }^{2}x$ using this identity, one can often transform a complex trigonometric equation into a simpler form, which might be a polynomial equation as seen in the exercise.

For example, in the solution provided, the Pythagorean identity is used to substitute ${\sin }^{2}x+{\cos }^{2}x$ with 1, reducing the equation to a quadratic form in terms of $\sin x$ and $\cos x$. This kind of manipulation is essential to simplify trigonometric equations, making them more solvable.

Trigonometric Equation Simplification

Trigonometric equation simplification is about reducing the complexity of an equation to a form where standard algebraic methods can then be employed. Simplification usually involves several steps, each making it easier to identify potential solutions. These steps include:

• Converting complex trigonometric functions into basic ones such as sine and cosine.
• Finding a common denominator to combine terms.
• Applying trigonometric identities like the Pythagorean identity to reduce the number of trigonometric functions in the equation.
• Rearranging the equation to isolate terms and solve as you would a polynomial or linear equation.

Throughout the simplification process, it is essential to remain aware of the domain and range of the functions involved as well as the restrictions placed on the variable, as seen with set S in the exercise.

In practice, as in our exercise, after rewriting the given trigonometric functions in terms of $\sin x$ and $\cos x$, combining terms over a common denominator, and using identities like the Pythagorean identity to simplify further, we reach a point where the equation resembles a standard quadratic. From here, we can utilize methods like factoring or the quadratic formula to solve for the variable, $x$. Each solution must then be checked to ensure it lies within the given domain and satisfies any additional conditions outlined in the problem, such as not being equal to $0,\pm \frac{\pi }{2}$.