Find cscθ Given sec θ = -2, sin θ >0

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In summary: If you solve for θ using the given equation and the criteria given, it turns out that θ falls into the second quadrant. In summary, the problem asks for the cscant of a value that falls into the second quadrant and the given equation yields that cscant as being 2√3/3.
  • #1

Homework Statement


Find cscθ given sec θ = -2 sin θ >0

Homework Equations


I do not know where to begin or what equations to use.

The Attempt at a Solution


I am assuming there is a typo in the question and that there should be a comma sec θ = -2, sin θ >0.

If so I got csc θ = 2√3 / 3
 
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  • #2
IntegralDerivative said:

Homework Statement


Find cscθ given sec θ = -2 sin θ >0

Homework Equations


I do not know where to begin or what equations to use.

The Attempt at a Solution


I do not know where to begin or what equations to use.
Why not? How is the secant defined? The cosecant? Those would be good places to start.
 
  • #3
I am assuming there is a typo in the question and that there should be a comma sec θ = -2, sin θ >0.

If so I got csc θ = 2√3 / 3
 
  • #4
IntegralDerivative said:
I am assuming there is a typo in the question and that there should be a comma sec θ = -2, sin θ >0.

If so I got csc θ = 2√3 / 3
Looks good.
 
  • #5
IntegralDerivative said:

Homework Statement


Find cscθ given sec θ = -2 sin θ >0

Homework Equations


I do not know where to begin or what equations to use.

The Attempt at a Solution


I am assuming there is a typo in the question and that there should be a comma sec θ = -2, sin θ >0.

If so I got csc θ = 2√3 / 3

The question as written (no comma) also makes perfectly good sense; it says that ##0 < \sec \theta = - 2 \sin \theta##, and you need to use the definition of ##\sec## to get a solvable equation. You can get ##\sin \theta## and ##\cos \theta##, then compute ##\csc \theta##, which does NOT equal ##2 \sqrt{3}/3 = 2/\sqrt{3}##.
 
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  • #6
IntegralDerivative said:

Homework Statement


Find cscθ given sec θ = -2 sin θ >0
As Ray points out, the problem as stated makes sense and can be solved. If you are currently studying trig equations in your course, then that's probably not a typo. On the other hand, if you are studying the basics of trigonometric functions early in your course, then it very well may be a typo, similar to what you have concluded.

The statement, ##\ \sec θ = -2 \sin θ >0 \,,\ ## is really two (or three) statements rolled into one.
  • ##\ \sec θ >0 \ ##
  • ##\ -2 \sin θ >0 \,,\ ## so that ##\ \sin θ <0 \ ##
  • ##\ \sec θ = -2 \sin θ ##
The first two tell you what quadrant θ is in.
 
  • #7
recast [itex] sec(\theta) [/itex] as [itex] 1/cos(\theta) [/itex] and the solution falls out in about 2 lines...
 

1. What is cscθ?

Cscθ is the reciprocal of the sine function, also known as the cosecant function. It is equal to 1/sinθ.

2. How do you find cscθ when given secθ = -2 and sinθ > 0?

First, we need to find the value of sinθ. Since we know that sinθ > 0, we can assume that θ is in either the first or second quadrant. Using the Pythagorean identity, sinθ = √(1 - cos²θ). Plugging in -2 for secθ, we get cosθ = -1/2. This gives us two possible values for θ: 60° or 300°. In either case, cscθ = 2/√3 or -2/√3, depending on which quadrant θ is in.

3. What is the relationship between secθ and cscθ?

Secθ and cscθ are inversely related. This means that if secθ = x, then cscθ = 1/x. In other words, when one is positive, the other is negative, and vice versa.

4. Can you use a calculator to find cscθ?

Yes, most scientific calculators have a csc or cosec button that allows you to find the cosecant of an angle directly. Make sure your calculator is in either degree or radian mode, depending on the unit of the given angle.

5. What is the range of values for cscθ?

The range of values for cscθ is all real numbers except for 0 and undefined values. This is because the denominator of the cosecant function (sinθ) cannot be 0, and there are no other restrictions on the possible values of sinθ.

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