Question

a point p(x,y) is shown on the unit circle corresponding to a real number t. find the values of the trigonometric functions at t. a. \\(\sin t = \frac{\sqrt{3}}{2}\\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize the denominator.) b. \\(\cos t = \frac{1}{2}\\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize the denominator.) c. \\(\tan t = \sqrt{3}\\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize the denominator.) d. \\(\csc t = \square\\) (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize the denominator.)

Explanation:

Step1: Recall csc definition

$\csc t = \frac{1}{\sin t}$

Step2: Substitute $\sin t$ value

$\csc t = \frac{1}{\frac{\sqrt{3}}{2}}$

Step3: Simplify the fraction

$\csc t = \frac{2}{\sqrt{3}}$

Step4: Rationalize the denominator

$\csc t = \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$

Answer:

$\frac{2\sqrt{3}}{3}$