Final answer:
To evaluate csc(-420°) without a calculator, find the equivalent angle within one period, which is -420° - 360° = -780°. Evaluate the sine function at -780°, which is sin(420°). Since it falls in the third quadrant, take the negative reciprocal to find csc(420°) = -2√3/3.
Step-by-step explanation:
To evaluate csc(-420°) without using a calculator, we need to find the reciprocal of the sine function. The sine function has a period of 360°. Therefore, we can find the equivalent angle within one period by subtracting or adding multiples of 360° to -420°. Since -420° is more than one full rotation, we can subtract 360° to get an equivalent angle of -420° - 360° = -780°. The angle -780° is coterminal with -420°.
Now, we can evaluate the reciprocal of the sine function at -780°. The sine function at -780° is sin(-780°) = sin(780°) = sin(780° - 360°) = sin(420°). Since the sine function is positive in the second and third quadrants, and the angle 420° falls in the third quadrant, we take the negative reciprocal to find csc(420°):
csc(420°) = -1/sin(420°).
We can use the unit circle or trigonometric ratios of special angles to find that sin(420°) = sin(60°) = √3/2. Therefore, csc(420°) = -1 / (√3/2) = -2/√3 = -2√3/3.