Math Problem Statement
Solution
The problem asks to find the values of for given that:
Step-by-step solution:
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Understanding the given information:
- : The cosecant function is the reciprocal of the sine function, so:
- : This means that must be in the first or fourth quadrant (since cosine is positive in these quadrants).
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Find the reference angle:
- To find the reference angle, take the inverse sine of the absolute value of :
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Determine the possible quadrants:
- Since is negative, must be in the third or fourth quadrant.
- However, since , must be in the fourth quadrant.
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Calculate the angle:
- In the fourth quadrant, the angle is given by:
Final answer:
Would you like more details or have any other questions?
Here are five related questions to expand on this concept:
- How would the solution change if ?
- What are the reciprocal identities for trigonometric functions?
- How can you solve for if given instead of ?
- What is the general solution for without the constraint ?
- How would you convert this problem to radians?
Tip: When using inverse trigonometric functions, always check which quadrant the angle belongs to based on the sign of the sine, cosine, or tangent functions.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Cosecant function
Cosine function
Reference angles
Formulas
csc θ = 1/sin θ
sin θ = 1/csc θ
Reference angle formula: θ_ref = arcsin(|sin θ|)
Theorems
Cosecant is the reciprocal of sine
Cosine positive in the first and fourth quadrants
Sine negative in the third and fourth quadrants
Suitable Grade Level
High school (Grades 10-12)