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How do you evaluate sine, cosine, tangent of #(4pi)/3# without using a calculator?

Answer 1

# - sqrt3/2 ; - 1/2 ; sqrt3#

Use trig table of special arcs and unit circle. On the unit circle, #sin ((4pi)/3) = - sin (pi/3) = - sqrt3/2# #cos ((4pi)/3) = - cos (pi/3) = - 1/2# #tan ((4pi)/3) = tan (pi/3) = sqrt3#

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Answer 2

Asher Friedman

To evaluate sine, cosine, and tangent of (4π)/3 without using a calculator, you can refer to the unit circle.

For (4π)/3:

The reference angle is π/3.
In the unit circle, π/3 corresponds to the point (1/2, √3/2).

Since (4π)/3 is in the second quadrant:

Sine is positive, so sin((4π)/3) = √3/2.
Cosine is negative, so cos((4π)/3) = -1/2.
Tangent is the ratio of sine to cosine, so tan((4π)/3) = (√3/2) / (-1/2) = -√3.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.