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Rotate the yellow dot to a location of 3 radians. After you rotate the angle, determine the value of csc 3, to the nearest hundredth.

Rotate the yellow dot to a location of 33 radians. After you rotate the angle, determine the value of csc⁡3 \csc 3 , to the nearest hundredth.

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Full solution

Q. Rotate the yellow dot to a location of 33 radians. After you rotate the angle, determine the value of csc⁡3 \csc 3 , to the nearest hundredth.
  1. Understand relationship csc(θ)sin⁡(θ)csc(\theta) \sin(\theta): Understand the relationship between csc(θ)csc(\theta) and sin⁡(θ)\sin(\theta). The cosecant function, csc(θ)csc(\theta), is the reciprocal of the sine function, sin⁡(θ)\sin(\theta). This means that csc(θ)=1sin⁡(θ)csc(\theta) = \frac{1}{\sin(\theta)}. To find csc(3)csc(3), we first need to find the value of sin⁡(3)\sin(3).
  2. Calculate value sin⁡(3)\sin(3): Calculate the value of sin⁡(3)\sin(3). Since we are working with radians, we can directly evaluate sin⁡(3)\sin(3). However, the sine function does not have a simple exact value for sin⁡(3)\sin(3). We will need to use a calculator to find the approximate value of sin⁡(3)\sin(3).
  3. Find reciprocal sin⁡(3)\sin(3) csc⁡(3)\csc(3): Find the reciprocal of sin⁡(3)\sin(3) to get csc⁡(3)\csc(3). Once we have the value of sin⁡(3)\sin(3), we can find csc⁡(3)\csc(3) by taking the reciprocal of sin⁡(3)\sin(3), as csc⁡(3)=1sin⁡(3)\csc(3) = \frac{1}{\sin(3)}.
  4. Use calculator find sin⁡(3)\sin(3) calculate csc⁡(3)\csc(3): Use a calculator to find sin⁡(3)\sin(3) and then calculate csc⁡(3)\csc(3). Using a calculator, we find that sin⁡(3)≈0.1411\sin(3) \approx 0.1411 (rounded to four decimal places). Now, we take the reciprocal of this value to find csc⁡(3)\csc(3). csc⁡(3)=1sin⁡(3)≈10.1411≈7.0869\csc(3) = \frac{1}{\sin(3)} \approx \frac{1}{0.1411} \approx 7.0869.
  5. Round value csc⁡(3)\csc(3): Round the value of csc⁡(3)\csc(3) to the nearest hundredth.\newlineRounding 7.08697.0869 to the nearest hundredth gives us 7.097.09.

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