Question 644062
Trigonometric functions are defined based on a set of x-y coordinates and the unit circle.
The unit circle (with radius=1) is centered at point O(0,0), the origin.
Angles are considered as clockwise or counterclockwise turns from a starting side placed over the positive x-axis, as ray OA in the figure below.
The starting side is always in that standard position. The starting side of an angle is always the positive x-axis.
The sweep that is the angle ends on a ray called the terminal side.
Counterclockwise turns are considered positive angles; clockwise turns are considered negative angles.
Angles can be measured in degrees. In trigonometry, they are often measured in radians.
{{{drawing(300,300,-1.7,1.7,-1.7,1.7,
grid(0),
circle(0,0,1),
blue(arrow(0,0,1.4,0)),green(arrow(0,0,1.1,1.1)),
blue(arrow(0,0,-1.4,0)),locate(0.05,-0.05,O),
locate(1.05,0.2,A),locate(-1.1,0.2,C),
locate(0.7,0.9,B)
)}}} If ray OA is turned counterclockwise until it coincides with ray OB, the angle swept can be measured as {{{45^o}}}.
The measure in radians is the distance traveled along the unit circle from point A to point B.
If ray OA is turned one whole turn counterclockwise, until it coincides with itself, the angle swept can be measured as {{{360^o}}}. In radians the measure would be {{{2pi}}}, because the distance traveled along the circle is the whole circumference of the unit circle, whose length is {{{2pi}}}
Doing the whole turn clockwise would sweep an angle of {{{-360^o}}} or, in radians {{{-2pi}}}.
An angle of {{{pi}}} would correspond to half a turn counterclockwise from OA to OC, while an angle of {{{-pi}}} would be half a turn clockwise from OA to OC. Both angles end in the same terminal side, and are called co-terminal. An infinite number of angle are co-terminal with those two. A {{{1&1/2}}} turn counterclockwise measures {{{3pi}}} is also co-terminal with {{{pi}}} and {{{-pi}}}, as would be a {{{2&1/2}}} clockwise turn , which would measure {{{-5pi}}}.
The trigonometric functions are based on the coordinates of the point where the terminal side crosses the unit circle. For the angle AOC measuring {{{-pi}}} (and all the angles co-terminal with that angle) that point is C(-1,0).
The sine of the angle is the y-coordinate of that point, so {{{sin(-pi)=0}}}.
The cosine of the angle is the x-coordinate of the point, so {{{cos(-pi)=-1}}}.
It is done the same way for any angle.
The other trigonometric functions (tangent, cotangent, secant and cosecant) can be calculated based on the sine and cosine of the same angle.
{{{tan(t)=sin(t)/cos(t)}}}
{{{cot(t)=cos(t)/sin(t)}}}
{{{sec(t)=1/cos(t)}}}
{{{csc(t)=1/sin(t)}}}
For {{{t=-pi}}}:
{{{tan(-pi)=0/(-1)=0}}}
{{{cot(-pi)}}} is undefined, because it would require dividing by {{{sin(-pi)=0}}}.
(I suppose they want you to write IMPOSSIBLE for an answer for {{{cot(-pi)}}}).
{{{sec(-pi)=1/(-1)=-1}}}
{{{csc(-pi)}}} is also undefined, because it would require dividing by {{{sin(-pi)=0}}}.
(I suppose they want you to write IMPOSSIBLE for that one too).