Can someone explain to me why the answer in this trigonometry question is what it is instead of something else?

Question

jojokiw3 · Answer

$$\sec (-\frac{ 19\pi }{ 6 })+\cos \frac{ 7\pi }{ 2 } - \sin \frac{ 10\pi }{ 3 } = \frac{ \sqrt{3} }{ 6 }$$ instead of $$= -\frac{ 7\sqrt{3} }{ 6 }$$?

jojokiw3 · Answer

$$-\frac{ \sqrt{3} }{ 6 }*$$

jim_thompson5910 · Answer

can you provide what you got for each little piece?

like what is the exact value of sec(-19pi/6) ?
what is the exact value of cos(7pi/2) ?
etc etc

jojokiw3 · Answer

Wait. Lemme think. It's that way because -sin 10pi/3 falls into the 3rd quadrant which makes it a double negative instead of one negative?

jojokiw3 · Answer

Because I got. -2/sqrt3 + 0 - sqrt3/2 whereas it should be -2/sqrt3 + 0 - (-sqrt3/2)?

jim_thompson5910 · Answer

sin(10pi/3) = -sqrt(3)/2

so

-sin(10pi/3) = -(-sqrt(3)/2) = sqrt(3)/2

you are correct

jojokiw3 · Answer

Ahh. I forgot to put the quadrants in consideration. Thanks!

jim_thompson5910 · Answer

you're welcome