Question

Explain how to do the following in good detail.
Without a calculator, determine the exact value of each expression.
Sin(pi/2) Cos(7pi/6) Tan(pi/6)

Answer 1

use a unit circle...radius of 1
cos refers to x distance from origin
sin refers to y distance from origin

pi/2 = 90 degrees, this is a quarter of a circle
starting at 3 o'clock, go 90 degrees to left you are at 12 o'clock
at this point you are vertical distance of 1 from origin
--> sin(pi/2) = 1

7pi/6 = 7*180/6 = 210 degrees
this is 180 + 30, so 30 degrees past half a circle
at this point, both x,y are negative
Now use geometry, a right triangle with hypotenuse of 1 and angles 30:60:90
side opposite 30 degree angle is 1/2
side opposite 60 degree angle is sqrt3/2
--> cos(7pi/6) = -sqrt3/2

pi/6 = 30 degrees
we already did this above, except now x,y are positive
sin(pi/6) = 1/2
cos(pi/6) = sqrt3/2
the tangent is sin/cos
--> tan(pi/6) = (1/2)/(sqrt3/2) = 1/sqrt3 = sqrt3/3