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Evaluate. Write your answer in simplified, rationalized form. Do not round.\newlinecsc⁡60∘=\csc 60^\circ = ______

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Csc, sec, and cot of special anglesright chevron icon

Full solution

Q. Evaluate. Write your answer in simplified, rationalized form. Do not round.\newlinecsc⁡60∘=\csc 60^\circ = ______
  1. Find sin⁡60°\sin 60°: To find the value of csc⁡60°\csc 60°, we first need to find the value of sin⁡60°\sin 60° and then take its reciprocal, since csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}.
  2. Calculate csc⁡60°\csc 60°: The value of sin⁡60°\sin 60° is known to be 3/2\sqrt{3}/2. Therefore, csc⁡60°=1/(sin⁡60°)=1/(3/2)\csc 60° = 1/(\sin 60°) = 1/(\sqrt{3}/2).
  3. Simplify 13/2\frac{1}{\sqrt{3}/2}: To simplify 13/2\frac{1}{\sqrt{3}/2}, we multiply both the numerator and the denominator by 22 to get rid of the fraction in the denominator. This gives us 23\frac{2}{\sqrt{3}}.
  4. Rationalize the denominator: To rationalize the denominator of 23\frac{2}{\sqrt{3}}, we multiply both the numerator and the denominator by 3\sqrt{3} to get 233\frac{2 \sqrt{3}}{3}.

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