Math Problem Statement

Find θ for 0° ≤ θ < 360° given that csc θ = -2.627 and cos θ > 0.

Solution

The problem asks to find the values of θ\theta for 0θ<3600^\circ \leq \theta < 360^\circ given that:

cscθ=2.627andcosθ>0\csc \theta = -2.627 \quad \text{and} \quad \cos \theta > 0

Step-by-step solution:

  1. Understanding the given information:

    • cscθ=2.627\csc \theta = -2.627: The cosecant function is the reciprocal of the sine function, so: cscθ=1sinθsinθ=1cscθ=12.6270.3807\csc \theta = \frac{1}{\sin \theta} \quad \Rightarrow \quad \sin \theta = \frac{1}{\csc \theta} = \frac{1}{-2.627} \approx -0.3807
    • cosθ>0\cos \theta > 0: This means that θ\theta must be in the first or fourth quadrant (since cosine is positive in these quadrants).
  2. Find the reference angle:

    • To find the reference angle, take the inverse sine of the absolute value of sinθ\sin \theta: θref=arcsin(0.3807)22.36\theta_{\text{ref}} = \arcsin(0.3807) \approx 22.36^\circ
  3. Determine the possible quadrants:

    • Since sinθ\sin \theta is negative, θ\theta must be in the third or fourth quadrant.
    • However, since cosθ>0\cos \theta > 0, θ\theta must be in the fourth quadrant.
  4. Calculate the angle:

    • In the fourth quadrant, the angle is given by: θ=360θref=36022.36337.64\theta = 360^\circ - \theta_{\text{ref}} = 360^\circ - 22.36^\circ \approx 337.64^\circ

Final answer:

θ338\theta \approx 338^\circ

Would you like more details or have any other questions?


Here are five related questions to expand on this concept:

  1. How would the solution change if cosθ<0\cos \theta < 0?
  2. What are the reciprocal identities for trigonometric functions?
  3. How can you solve for θ\theta if given cotθ\cot \theta instead of cscθ\csc \theta?
  4. What is the general solution for sinθ=0.3807\sin \theta = -0.3807 without the constraint 0θ<3600^\circ \leq \theta < 360^\circ?
  5. 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)