Solving: secx=-5.2, 0≤x≤2π: Why Only 1 Answer?

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In summary, the answer to why there is only one solution to the equation secx=-5.2 lies in the definition of secant in trigonometry. The domain is limited to 0≤x≤2π because the function secant has a period of 2π. The inverse function cannot be used to find more solutions because it is not defined for values outside of -1 to 1, and there are ways to approximate a solution using numerical methods. However, it is not possible to have more than one solution to this equation, as the range of secant is restricted to values of -1 or less and -5.2 is not a possible value for secant.
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Destrio
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Solve: secx=-5.2, 0≤x≤2π

I'm getting 2 answers:
1.76, 4.52

The answer key is telling me there is only one
1.76

How do I only get 1 answer from this?

Thanks,
 
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  • #2
damn, wrong forum
sorry, I'll repost
 
  • #3


I understand your confusion about the number of solutions for this equation. However, it is important to remember that when solving equations, we must consider the domain of the variable. In this case, the given domain is 0≤x≤2π, which means that x can only take on values between 0 and 2π.

When we solve the equation, we get two possible values for x: 1.76 and 4.52. However, only one of these values falls within the given domain, which is 1.76. Therefore, for this specific problem, there is only one solution that satisfies both the equation and the given domain.

In other cases, where the domain is not specified, we may get multiple solutions for an equation. It is important to always check the domain and make sure that the solutions obtained fall within it. I hope this explanation helps clarify why there is only one solution for this particular problem. Keep up the great work in solving equations!
 

Related to Solving: secx=-5.2, 0≤x≤2π: Why Only 1 Answer?

1. Why is there only one answer to the equation secx=-5.2?

The answer to this question lies in the definition of secant (sec) in trigonometry. Secant is the reciprocal of cosine (cos) and is defined as the ratio of the hypotenuse to the adjacent side of a right triangle. Since cosine has a range of -1 to 1, secant can only have a value of -1 or less. In this case, -5.2 is not a possible value for secant, thus there is no solution.

2. Why is the domain limited to 0≤x≤2π?

The domain of a trigonometric equation is typically limited to one period of the function. In this case, the function secant has a period of 2π, meaning that it repeats itself every 2π units. Therefore, restricting the domain to 0≤x≤2π ensures that we are only considering one full cycle of the function.

3. Can't we use the inverse function to find more solutions?

The inverse function of secant is not defined for all values of secant. It is only defined for values between -1 and 1. Since -5.2 is not a possible value for secant, we cannot use the inverse function to find more solutions.

4. Is there a way to approximate a solution?

Yes, there are ways to approximate a solution using numerical methods such as the Newton-Raphson method or the bisection method. These methods involve repeatedly evaluating the function at different values until you get close enough to the desired solution.

5. Is it possible to have more than one solution to this equation?

No, there is only one solution to this equation. As mentioned before, the range of secant is restricted to values of -1 or less, and -5.2 is not a possible value for secant. Therefore, there can only be one solution to the equation.

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