There are six trig functions which all describe relationships on a right triangle (for now)

sin𝜭 = opposite_side/hypotenuse

cos𝜭 = adjacent_side/hypotenuse

tan𝜭 = opposite_side/adjacent_side

You can remember these three with SOH-CAH-TOA

The next three are reciprocals of the first three

  • Sine -> cosecant
  • Cosine -> secant
  • Tangent -> cotangent
csc𝜭 = hypotenuse/opposite_side

sec𝜭 = hypotenuse/adjacent_side

cot𝜭 = adjacent_side/opposite_side

Because of this reciprocal relationship, you can convert between trig functions:

EX: sec𝜭 = 1/cos𝜭

Signs by quadrent

Keep in mind that triangles can be in any of the four quadrents, and trig identities will have different signs depending on the quadrent.

If you know the quadrent of a triangle and one trig function, you can find the other five.

    I    II    III    IV
sin +     +     -      -
csc +     +     -      -
cos +     -     -      +
sec +     -     -      +
tan +     -     +      -
cot +     -     +      -
  1. Use a pythagorean identity to find another function
  2. Use quotient identities/reciprocal identities to find the remaining functions

Undefined

Trig functions can be undefined at one of the quadrantel angles (exactly on an axis of a unit circle)

This is because you end up with x = y/0

radians of unit circle

      sin  csc  cos  sec  tan  cot
0     0    und  1    1    0    und
pi/2  1    1    0    und  und  0
pi    0    und  -1   -1   0    und
3pi/2 -1   -1   0    und  und  0
2pi   0    und  1    1    0    und