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The variable \( t \) is a real number and \( P = \left(\frac{27}{30}, \frac{11}{30}\right) \) is the point on the unit circle that corresponds to \( t \). Find the exact values of the six trigonometric functions of \( t \).

1. \(\sin t\) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
2. \(\cos t\) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
3. \(\tan t\) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
4. \(\csc t\) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
5. \(\sec t\) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
6. \(\cot t\) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

Answer :

Final answer:

To find the exact values of the trigonometric functions of t on the unit circle, we can use the coordinates of the point (cos(t), sin(t)). By applying the definitions of the trigonometric functions, we can simplify the expressions and obtain the exact values.

Explanation:

To find the exact values of the six trigonometric functions of t, we can use the point on the unit circle that corresponds to t. The coordinates of this point are given as (cos(t), sin(t)). Since the point lies on the unit circle, the distance from the origin to the point is 1. Therefore, we have:




  2. sint = sin(t) = Opposite / Hypotenuse = y-coordinate / 1 = sin(t)

  4. cost = cos(t) = Adjacent / Hypotenuse = x-coordinate / 1 = cos(t)

  6. tant = tan(t) = Opposite / Adjacent = sin(t) / cos(t) = tan(t)

  8. csct = csc(t) = 1 / sin(t) = 1 / sin(t)

  10. sect = sec(t) = 1 / cos(t) = 1 / cos(t)

  12. cott = cot(t) = cos(t) / sin(t) = cos(t) / sin(t)

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