Question

For each t value below, draw a sketch of it; state the Quadrant its terminal point lies in; and then find the value of the six trigonometric functions of t.

a. t=13π/6
b. t=-2π/3
c. t=-5π/4

Answer 1

The angles given are analyzed and expressed in terms of positive coterminal angles, then the quadrant each terminal point lies in and the values of six trigonometric functions are determined for each.

Step-by-step explanation:

To solve the question, we must analyze the given values of t and determine their trigonometric functions, as well as the quadrant in which their terminal points lie.

1. t = 13π/6: Simplify the fraction by dividing the numerator by the denominator. Since 13/6 is greater than 2, we can subtract 2π (which is the same as 12π/6) to get a coterminal angle, t = π/6. This coterminal angle lies in Quadrant I, where all trigonometric functions are positive. The trigonometric functions are as follows: sin(t) = 1/2, cos(t) = √3/2, tan(t) = 1/√3, csc(t) = 2, sec(t) = 2/√3, and cot(t) = √3.
2. t = -2π/3: This angle is negative, indicating clockwise rotation. By adding 2π, we find a coterminal angle that lies in Quadrant III, where sin and cos are negative and tan is positive. The trigonometric functions are: sin(t) = -√3/2, cos(t) = -1/2, tan(t) = √3, csc(t) = -2/√3, sec(t) = -2, and cot(t) = 1/√3.
3. t = -5π/4: Again, a negative angle indicating clockwise rotation. By adding 2π, we find a coterminal angle in Quadrant III. The trigonometric functions are: sin(t) = -√2/2, cos(t) = -√2/2, tan(t) = 1, csc(t) = -√2, sec(t) = -√2, and cot(t) = 1.

Remember, expressing a negative angle in terms of a positive coterminal angle makes it easier to determine the quadrant and function values.