If [tex]$\csc \theta=\frac{1}{\sin \theta}$[/tex] and [tex]$\sin \theta=\frac{1}{\sqrt{2}}$[/tex], for the angle 450 what is [tex]$\operatorname{Csc} \theta$[/tex]?
A. [tex]$\frac{2}{\sqrt{22}}$[/tex]
B. [tex]$\frac{2 \sqrt{2}}{2}$[/tex]
C. [tex]$\frac{\sqrt{2}}{1}$[/tex]
Answer (2)
Find the value of sin ( 45 0 ∘ ) which is equal to sin ( 9 0 ∘ ) = 1 .
Use the identity csc θ = s i n θ 1 to find csc ( 45 0 ∘ ) = 1 1 = 1 .
Compare the result with the given options. None of the options equal 1.
The correct value for csc ( 45 0 ∘ ) is 1, but none of the options match. There might be a typo in the question.
Assuming the question intended to ask for the cosecant of an angle whose sine is 2 1 , the answer would be 2 . But since the question explicitly asks for csc ( 45 0 ∘ ) , the answer is 1 .
Explanation
Understanding the Problem We are given that csc θ = s i n θ 1 and sin θ = 2 1 . We need to find the value of csc θ for the angle 450 degrees.
Finding the Sine of 450 degrees First, we need to find the value of sin ( 45 0 ∘ ) . Since the sine function has a period of 36 0 ∘ , we can subtract 36 0 ∘ from 45 0 ∘ without changing the sine value: sin ( 45 0 ∘ ) = sin ( 45 0 ∘ − 36 0 ∘ ) = sin ( 9 0 ∘ )
Sine of 90 degrees We know that sin ( 9 0 ∘ ) = 1 . Therefore, sin ( 45 0 ∘ ) = 1 .
Finding the Cosecant of 450 degrees Now, we can find the value of csc ( 45 0 ∘ ) using the identity csc θ = s i n θ 1 : csc ( 45 0 ∘ ) = sin ( 45 0 ∘ ) 1 = 1 1 = 1
Comparing with the given options So, csc ( 45 0 ∘ ) = 1 . Now we need to compare this result with the given options to select the correct answer. We can rewrite the options to see which one is equal to 1. A. 22 2 ≈ 0.426 , which is not equal to 1. B. 2 2 2 = 2 ≈ 1.414 , which is not equal to 1. C. 1 2 = 2 ≈ 1.414 , which is not equal to 1. However, based on our calculations, csc ( 45 0 ∘ ) = 1 . None of the given options match our result. There seems to be an error in the problem statement or the given options. The correct value for csc ( 45 0 ∘ ) is 1.
Final Answer and Conclusion Since csc ( 45 0 ∘ ) = 1 , we need to find an equivalent expression among the options. However, none of the options equal 1. Let's re-evaluate the problem statement. We are given that sin θ = 2 1 . This information seems irrelevant since we are asked to find csc ( 45 0 ∘ ) . If we were asked to find csc θ when sin θ = 2 1 , then csc θ = s i n θ 1 = 2 1 1 = 2 . In this case, option C, 1 2 , would be the correct answer. However, the question asks for csc ( 45 0 ∘ ) , which is 1. There might be a typo in the question. Assuming the question intended to ask for the cosecant of an angle whose sine is 2 1 , the answer would be 2 . But since the question explicitly asks for csc ( 45 0 ∘ ) , the answer is 1.
Examples
Cosecant, as the reciprocal of sine, helps in fields like navigation and astronomy. For instance, when tracking a satellite's orbit, cosecant can be used to determine the satellite's altitude relative to an observer on Earth, given the angle of elevation. This is crucial for maintaining communication links and predicting satellite visibility. Understanding trigonometric functions like cosecant allows for precise calculations in various real-world applications involving angles and distances.
The cosecant of 450 degrees is calculated as csc ( 45 0 o ) = 1 . None of the provided answer options match this result, which suggests that there might be an error in the question. Thus, the answers listed do not represent the correct value of cosecant for the specified angle.
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