Question

1) Evaluate (if possible) the sine, cosine, and tangent of the angles without a calculator.

a) 10pi/3

Sin = -sqrt 3/2
Cos = -1/2
Tan = sqrt 3

Are these correct ? I do not understand when to make them negative.

b) 17pi/3

Sin = -sqrt 3/2
Cos = 1/2
Tan = - sqrt 3

Are these correct?

Answer 1

your answers are correct, (but I have a feeling you used your calculator and then decided what radical your answer was)

Some students find it easier to "think" in degrees than radians
10pi/3 radians = 600º
you have to change that to a co-terminal angle, that is, an angle in the first rotation landing in the same position.
To do that subtract multiples of 360 until you have an answer less than 360
600-360 = 240

Now ask yourself, "Where is 240º ?"
- in the third quadrant.
In III, the sine is -, the cosine is +, and the tangent is +

Now ask yourself, "What is the angle in standard position equal to 240º ?"
- that is, how far to the x-axis ? -- 60º
You should know the sine, cosine and tangents of 60º
Using the proper sines for III
sin 10pi/3 = sin 600º = sin 240º = -sin 60º = -√3/2

etc.

Practice with a few, after a while this is very quick and easy.